SARS Cov-2 infectiousness decays exponentially, with the time constant a function of temperature and humidity. A two-parameter calculator: https://www.dhs.gov/science-and-technology/sars-calculator
Sounds like wetting the towel first would be the way to go.
Also, the humidity factor seems counter intuitive to me on that site. I thought the virus survived longer in the air in damp spaces, while sunlight dried out the virus rendering it inert.