Here's the thing. That phrase "burden of proof" is used by those building fake authority to build up an audience and sell pseudointellectual content to their userbase... the likes of Sam Harris, Scott Adams, Neil d. Tyson etc. are known to do that and it seems to be used in those crowds. It seems like a great phrase when they seem to use it when talking to people they already agree with and make other people look silly (video title: "Sam Harris owns X") and it may be a hit in that crowd where the audience is clueless about it... but if you take it outside, people will point out to you that it's silly.

In forums like this, when it comes to new things, people are trying to make sense of what is happening by sharing their opinions and what they are basing them on... in hopes of getting either supporting or opposing views and letting that collective opinion sharing guide themselves and others into what might really be happening. We're not defending out scientific paper nor are we building up fake authority to scam people.