I know of no scientist who claims that they have proof of the theories that they have about the origins of the universe, although there is constant research on these theories, and they are in the public domain. There is no-one trying to force these ideas on others either. The big bang is a theory, regardless of what TV shows try and make out to the contrary. I have, however, heard many christians claim that god created the universe, without any proof whatsoever.
"Nobody is obligated to prove or disprove anything, unless they are trying to coerc or force it on others, in which case, the burden of proof is on them."
Nobody can expect someone to believe a claim without proof of the claim, if one makes the claim, one has the burden of proof to provide evidence to support the claim, particularly if the one making the claim wants to be taken seriously. Without evidence the claim is baseless, and cannot be called truth.
A claim that does not have evidence to back it up, is merely an opinion, and worth no more or less than any other opinion.