As of the proof of the value of BB(5), there are three candidates for the next Busy Beaver value (and corresponding machine) to find:

BB(6), aka BB(6,2) is the 6-state 2-symbol machine.

BB(3,3) is the 3-state 3-symbol machine.

BB(2,5) is the 2-state 5-symbol machine.

Any proof of the machine for more states or symbols would imply a proof for one or more of the above problems sizes. If multiple problem sizes are proven simultaneously, those options will resolve in equal proportion.

For this question all that is required is a proof as to which machine achieves the BB(x,y) bound in question. An exact value for S(x,y) or Σ(x,y) is not required. An explicit machine is required, along with a proof that it halts and a proof that no other longer-running machine halts. See also /EvanDaniel/will-the-bb6-machine-be-known-by-20 , which uses the same criteria.

A formal announcement by the bbchallenge.org site, peer reviewed papers, or statements by prominent computational complexity experts will be required to resolve this question.