I'm going to object to this bandying about of the phrase 'burden of proof'. There is no universal concept of 'burden of proof' that is automatically generated for every disagreement. You cannot say that the burden of proof lies with them, as the meaning of proof and burden of proof differs in each logical system. As it turns out, people are not working on the same logical system.
For example, take
goldbach's conjecture. Some might consider it 'proved', since it has been verified a lot of numbers (up to 1.609 × 10
18). Others might consider it not proved, since it has not been proved for all numbers.
The concept of burden of proof works in a similar way. Some might consider the burden of proof to lie on those who do not support it to find a counterexample, while others might consider the burden of proof to lie on those who support it to find a proof for all numbers.
Given that this is a mathematical problem, and the nature of mathematical proofs, some might consider it a bad analogy, as the mathematical definition of proof does not consider how many instances it has been verified in (except for a finite domain of discourse, in which case exhaustion may apply). Instead, consider an analogous physics experiment - you chuck a ball into the air and time how long it takes to hit the ground. You do this 1.6 sextillion times, and every single time it hits the ground at the time predicted by a certain model (within experimental error). A great deal of people would consider the model proved in this situation.
TLDR; make sure you are specific in what logical system you are using and what you mean by proof before you tell other people to provide you with one.