  Pudl\'ak [Pud17] lists several major conjectures from the field of proof
complexity and asks for oracles that separate corresponding relativized
conjectures. Among these conjectures are:
  - $\mathsf{DisjNP}$: The class of all disjoint NP-pairs does not have
many-one complete elements.
  - $\mathsf{SAT}$: NP does not contain many-one complete sets that have
P-optimal proof systems.
  - $\mathsf{UP}$: UP does not have many-one complete problems.
  - $\mathsf{NP}\cap\mathsf{coNP}$: $\text{NP}\cap\text{coNP}$ does not have
many-one complete problems.
  As one answer to this question, we construct an oracle relative to which
$\mathsf{DisjNP}$, $\neg \mathsf{SAT}$, $\mathsf{UP}$, and
$\mathsf{NP}\cap\mathsf{coNP}$ hold, i.e., there is no relativizable proof for
the implication $\mathsf{DisjNP}\wedge \mathsf{UP}\wedge
\mathsf{NP}\cap\mathsf{coNP}\Rightarrow\mathsf{SAT}$. In particular, regarding
the conjectures by Pudl\'ak this extends a result by Khaniki [Kha19].
