I just watched a documentary on Ad-36, which doesn't make me an expert, but presumably I now know more than many of the previous people who've commented here:

They checked a pair of twins who had very different body weights (which is said to be unusual, since identical twins tend to maintain nearly identical weights). The fatter twin tested positive for antibodies for the adenovirus; the skinny (normal) twin tested negative, meaning she had not been exposed. That seems like a pretty positive confirmation.

The documentary suggested that fat people are not contagious... the virus comes along, modifies a bunch of "pre-fat" cells turning them into regular fat cells, and doesn't stick around. The damage is done, and probably cannot be reversed.

They tested a bunch of people in the UK; a substantial portion of the obese people did test positive for exposure to the adenovirus. The lean people that were tested as a control sample were all negative.

The virus can't be blamed for all the overweight people, but even if it's only 1/4, that's a heck of a lot of unwanted jiggling cellulite, Type 2 diabetes, heart disease, humiliation and frustration that could be easily preventable if there were a vaccine against these adenoviruses.