HIV is still transmissible even with HAART therapy, so I'm not sure how it's relevant to the discussion of a cure.
Transmissibility also relates to transmission vectors: eg vaginal or anal intercourse, sharing needles with intravenous drugs, and infected-mother-to-child routes (eg placental and/or breastfeeding).
In particular the "Undetectable = Untransmissible" claim is (as I understand it) a claim specifically about transmission through vaginal and anal intercourse.
> Sexual transmission of HIV according to viral load and antiretroviral therapy: systematic review and meta-analysis
> Results: We identified 11 cohorts reporting on 5021 heterosexual couples and 461 HIV-transmission events. The rate of transmission overall from ART-treated patients was 0.46 (95% CI 0.19–1.09) per 100 person-years, based on five events. The transmission rate from a seropositive partner with viral load below 400 copies/ml on ART, based on two studies, was zero with an upper 97.5% confidence limit of 1.27 per 100 person- years, and 0.16 (95% CI 0.02–1.13) per 100 person-years if not on ART, based on five studies and one event. There were insufficient data to calculate rates according to the presence or absence of sexually transmitted infections, condom use, or vaginal or anal intercourse.
> Conclusion: Studies of heterosexual discordant couples observed no transmission in patients treated with ART and with viral load below 400copies/ml, but data were compatible with one transmission per 79 person-years. Further studies are needed to better define the risk of HIV transmission from patients on ART.
...
> Probability of HIV-1 transmission per coital act in monogamous, heterosexual, HIV-1-discordant couples in Rakai, Uganda
> RESULTS The mean frequency of intercourse was 8.9 per month, which declined with age and HIV-1 viral load. Members of couples reported similar frequencies of intercourse. The overall unadjusted probability of HIV-1 transmission per coital act was 0.0011 (95% CI 0.0008-0.0015). Transmission probabilities increased from 0.0001 per act at viral loads of less than 1700 copies/mL to 0.0023 per act at 38 500 copies/mL or more (p=0.002), and were 0.0041 with genital ulceration versus 0.0011 without (p=0.02). Transmission probabilities per act did not differ significantly by HIV-1 subtypes A and D, sex, STDs, or symptoms of discharge or dysuria in the HIV-1-positive partner. INTERPRETATION Higher viral load and genital ulceration are the main determinants of HIV-1 transmission per coital act in this Ugandan population.
Edit: While my comment came from caution due to historical human hubris, it appears I am wrong:
“According to the Centers for Disease Control and Prevention (CDC), a person with an undetectable viral load has “effectively no risk” of sexually transmitting HIV. In 2016, the Prevention Access Campaign launched the U=U, or Undetectable = Untransmittable, campaign.”
Here [1] is for instance an article referencing this claim, describing it as "based on strong scientific evidence" that was peer reviewed published by the journal of the American Medical Association. (EDIT: it's important to add the caveat, which is that this claim is about sexual transmission only, and does not cover pregnant-mother-to-child transmission or intravenous needle sharing).
Also your comment now refers to "HPV" which is the name of a different disease.
[1] https://jamanetwork.com/journals/jama/article-abstract/27209...
Too often we think we’re smarter than we are, and the consequences can be damning. Hence why I said its a bold claim that undetectable means it can’t be transmitted. Are we absolutely sure? Because if we’re not, someone could acquire a life long chronic disease that would otherwise have been preventable. But the CDC says undetectable says no transmission; I hope they’re right in 100% of transmission scenarios.
https://www.healthline.com/health/hiv-aids/undetectable-vira...