Consequences of a $p$-optimal proof system for $\operatorname{TAUT}$
See the Wikipedia article for propositional proof complexity. p-simulation is similar to reductions between complexity classes. Existence of a p-optimal proof system is a well-known open problem in proof complexity.

It is statement 7 in Krajicek and Pudlak 1989. See page 1066 and 1067 for a list of statements implied/implying it. There are more statements that you can find by Googling.

There are experts who conjecture that extended Frege is an p-optimal proof system. Existence of a p-optimal proof system is a weaker statement than existence of a super proof system (which is equivalent to $\mathsf{NP}=\mathsf{coNP}$). 

Jan Krajicek and P. Pudlak, "Propositional Proof Systems, the Consistency of First Order Theories and the Complexity of Computations", J. Symbolic Logic, 54(3), (1989), pp. 1063-1079.

Yijia Chen and Jörg Flum, "On optimal proof systems and logics for PTIME", 2010.

Zenon Sadowski, "On an optimal propositional proof system and the structure of easy subsets of TAUT", 2002.