I wouldn't say the BOP falls on whoever cares more. Some one claims; "Chapstick is bad for you." The second person might respond with, "I don't believe that." Who is BOP on? The one who made the claim, not the one who doesn't believe the claim.
You should still be doing your due diligence to disprove a false statement ... Whether the burden is implicit or whatever. People make blanket statements like that all the time. They aren't automatically disproved by saying "you didn't provide good sources etc. The statement is still out there, still able to do damage as a lie until you (the person who cares to snuff it out) bears that burden and disproves it. It's a little dog-eat-dog, but I think thats what it takes for truth to be told.
Also I dont think you can come to support a rivals claim by default, just because the first guy failed BOP. No BOP proven is a tie as far as im concerned. The burden is on both.