  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
$\mathsf{TAUT}$, assuming the collapse of $\cal C$ and $\sf N{\cal C}$ (the
nondeterministic version of $\cal C$) for some new classes $\cal C$ and also
prove new conditional independence results for strong theories, assuming
nonexistence of p-optimal proof systems.
  3. We construct two new oracles ${\cal V}$ and ${\cal W}$. 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
$\mathsf{TAUT}$ and existence of a complete problem for $\mathsf{TFNP}$ are
independent of each other in relativized worlds which was not known before.
