New relations and separations of conjectures about incompleteness in the finite domain

Our main results are in the following three sections:

1. We prove new relations between proof complexity conjectures that are discussed in \cite{pu18}.
2. We investigate the existence of p-optimal proof systems for $\mathsf{T}\mathsf{A}\mathsf{U}\mathsf{T}$, assuming the collapse of $\mathcal{C}$ and $\mathsf{N}\mathcal{C}$ (the nondeterministic version of $\mathcal{C}$) for some new classes $\mathcal{C}$ and also prove new conditional independence results for strong theories, assuming nonexistence of p-optimal proof systems.
3. We construct two new oracles $\mathcal{V}$ and $\mathcal{W}$. These two oracles imply several new separations of proof complexity conjectures in relativized worlds. Among them, we prove that existence of a p-optimal proof system for $\mathsf{T}\mathsf{A}\mathsf{U}\mathsf{T}$ and existence of a complete problem for $\mathsf{T}\mathsf{F}\mathsf{N}\mathsf{P}$ are independent of each other in relativized worlds which was not known before.