"I have a request."
*brief pause*
"Will you make eye contact with me for 60 seconds?"
I like many things about this ask:
1. It's not as abrupt as simply saying "Will you make eye contact with me for 60 seconds?" with no prior intro.
2. It doesn't frame the request as being part of an experiment. I prefer not to refer to this as an experiment, because it makes the action seem less personal and less intimate. Saying it is an experiment also hits at my purpose, and I prefer to not discuss why I am doing this until after they agreed to do it with me, or not.
3. Although it's not a goal of this project to get people to say "yes" to making eye contact, saying "I have a request." first, with a pause, then making the request, puts the other person into the mindset to hear and respond to a request, which I think will make them more likely to say "yes" to it. Without this setup, they may be needlessly surprised by the unusual and bold request.