Study of TP-04 in Participants With Papulopustular Rosacea

Inclusion criteria: 1. Participant is willing to participate and is capable of giving informed consent. Note: Consent must be obtained prior to any study-related procedures. 2. Male or female participant aged 18 to 59 years, inclusive, at the time of consent. 3. Participant has at least a 6-month history of PPR at the screening visit (information obtained from medical chart or participant's physician, or directly from the participant). 4. Participant has moderate or severe PPR, as defined by an IGA score of 3 (moderate) or 4 (severe) at the screening and Day 1 visits. 5. Participant has 20 to 70 inflammatory lesions (papules and/or pustules) and no more than 2 nodules (defined as a lesion ˃ 5 mm in diameter) on the face at the screening and Day 1 visits. 6. Participant has moderate or severe persistent erythema associated with PPR, as defined by a CEA score of 3 (moderate) or 4 (severe) at the screening and Day 1 visits. 7. Participant has SSSB1 \> 5 D/cm² or SSSB2 \> 10 D/cm² at the screening and Day 1 visits. 8. Female participant of childbearing potential has had a negative serum pregnancy test at screening and a negative urine pregnancy test at Day 1. 9. For female participant of childbearing potential involved in any sexual intercourse that could lead to pregnancy: the participant must agree to use a highly effective contraceptive method from at least 4 weeks prior to Day 1 until at least 6 months after the last study product application. Highly effective contraceptive methods include hormonal contraceptives (eg, combined oral contraceptive, patch, vaginal ring, injectable, or implant), intrauterine devices or intrauterine systems, vasectomized partner(s) (provided his vasectomy was performed ≥ 4 months prior to screening), tubal ligation or double barrier methods of contraception (eg, male condom with cervical cap, male condom with diaphragm, and male condom with contraceptive sponge) in conjunction with spermicide. 10. For male participant involved in any sexual intercourse that could lead to pregnancy, participant must agree to use one of the highly effective contraceptive methods listed in Inclusion Criterion #9, from Day 1 until at least 6 months after the last study product application. If the female partner of a male participant use any of the hormonal contraceptive methods listed above, this contraceptive method should be used by the female partner from at least 4 weeks before Day 1 until at least 6 months after the last study product application. 11. For participant who uses makeup, facial moisturizers, creams, lotions, cleansers, and/or sunscreens, participant has used the same product brands/types for a minimum period of 2 weeks prior to Day 1, agrees not to change brand/type or frequency of use throughout the study, and agrees not to use makeup, facial moisturizers, creams, lotions, cleansers, and/or sunscreens prior to study visits. Participants will be instructed not to apply these products on the treated areas within approximately 30 minutes before and after study product application. 12. Participant is willing to limit or avoid known personal triggers of rosacea (eg, spicy foods, consumption of alcoholic beverages, extended intentional sun exposure, tanning beds, sauna, etc) for the duration of the study. 13. Participants must be willing to comply with all study procedures and must be available for the duration of the study. Exclusion criteria: 1. Participant is a female who is breastfeeding, pregnant, or who is planning to become pregnant during the study. 2. Participant has rosacea conglobata, rosacea fulminans, perioral dermatitis, facial erythrosis other than rosacea, corticosteroid-induced rosacea, facial keratosis pilaris, facial seborrheic dermatitis, acute lupus erythematosus, chronic recurring facial acne vulgaris, isolated rhinophyma, or plaque- like facial edema or with ocular rosacea (blepharitis, keratitis) requiring or likely to require systemic treatment. 3. Participant has a history of skin disease, presence of a skin condition, scarring, excessive facial hair, tattoos, or other facial characteristics (eg, actinic damage) that could, in the opinion of the investigator, interfere with study assessments. 4. Participant has any clinically significant medical condition or physical/laboratory/ECG/vital signs abnormality that would, in the opinion of the investigator, put the participant at undue risk or interfere with interpretation of study results. 5. Participant has a history of cancer or lymphoproliferative disease within 5 years prior to Day 1. Participants with successfully treated nonmetastatic cutaneous squamous cell or basal cell carcinoma and/or localized carcinoma in situ of the cervix are not to be excluded. 6. Participant has a known history of chronic infectious disease (eg, hepatitis B, hepatitis C, or human immunodeficiency virus \[HIV\]). 7. Participant has used oral retinoids (eg, isotretinoin) within 52 weeks prior to Day 1 or high-dose vitamin A (\> 10,000 IU/day) within 26 weeks prior to Day 1. 8. Participant has received any marketed or investigational biological agent within 12 weeks or 5 half-lives (whichever is longer) prior to Day 1. 9. Participant has received hormonal therapy that is not on a stable dose and frequency for at least 12 weeks before Day 1 or that is not maintained throughout the study. 10. Participant has received photodynamic therapy, phototherapy with blue or red light, or laser therapy to the face within 8 weeks prior to Day 1. 11. Participant had a major surgery within 8 weeks prior to Day 1 or has a major surgery planned during the study. 12. Participant has used systemic immunosuppressants (eg, steroids, steroid injections, cyclosporine, methotrexate, mycophenolate mofetate) within 8 weeks prior to Day 1. 13. Participant had a facial procedure (eg, chemical peel, microdermabrasion) within 8 weeks prior to Day 1. 14. Participant has used systemic products that could affect PPR within 4 weeks prior to Day 1 (eg, oral antibiotics, ivermectin). 15. Participant is currently receiving a nonbiological investigational product or device or has received one within 4 weeks prior to Day 1. 16. Participant has used any narrow therapeutic index (NTI) drug metabolized by cytochrome P450 (CYP)2C8, CYP2C19, CYP2C9, or CYP2D6 within 4 weeks prior to Day 1 or may require using any NTI drug metabolized by CYP2C8, CYP2C19, CYP2C9, or CYP2D6 within 8 weeks after the last study product application. 17. Participant is exposed to excessive sunlight (eg, working outside), is planning a trip to a sunny climate, or has used tanning booths within 4 weeks prior to Day 1 or is not willing to minimize exposure to natural and artificial sunlight or weather extremes (wind or cold) during the study. 18. Participant has used topical products that could affect PPR within 2 weeks prior to Day 1 (eg, metronidazole, azealeic acid, topical minocycline, sulfacetamide, resorcinol, clindamycin, topical ivermectin, tea tree oil, dapsone, cannabidiol \[CBD\]-containing products, topical calcineurin inhibitors, benzyl peroxide, topical retinoid/retinol, alpha or beta hydroxy-acids, phosphodiesterase 4 \[PDE4\] inhibitors, topical Janus kinase \[JAK\] inhibitors, oxymetazoline, bromonidine). 19. Participant has used on the face an over-the-counter (OTC) or prescription topical medication for rosacea, including benzoyl peroxide, topical anti-inflammatory medications, corticosteroids, salicylic acid, α-hydroxy/glycolic, or antibacterial/antiseptic soap or wash within 2 weeks prior to Day 1. 20. Participant has a known or suspected allergy to Lotilaner Gel, 2.0% (TP-04) or any component of the investigational product.