Spreading through a damp towel?

Scenario: A person with HR HPV (like 16, which apparently can survive on fomites for 7 days) touches himself while peeing, washes hands and dries them in a towel. The next person washes hands, dries them in a towel and then touches himself while peeing. Is the second person at risk of contracting HPV that way?

I know a person who contracted HPV after getting the first shot of Gardasil :(