Time-traveling is a contagious meme, and a fragile one.
That is the funny thing. Let me tell you how I see it.
Anybody could present himself/herself as a time-traveler. He needs just some imagination and nothing else. There are little tricks like Basiago Gettysburg picture, which will put some spice and realism to the hoax.
My idea is that the time-travel meme will die by excess. The same happened with the visits to other planets in the 50's. After some time, there were too much travels and the thing lost interest and credibility. It didn't sell anymore.
So, back into time-travel mythology, in a couple of months there will be a thousand Basiago-like time-travelers. No evidence needed.
The pretender will invent time-visits to old worlds and even some romances with Cleopatra or Helen of Troy.
After sometime the whole meme will die a natural dead.
Something more: when we talk of memes like Basiago-like time-jumping. We are talking about things that only exist inside the UFO and paranormal subculture. Be aware that in everyday world nobody knows or cares about Andy Basiago's time and space jumps.