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Pudl\'ak [Pud17] lists several major conjectures from the field of proofcomplexity and asks for oracles that separate corresponding relativizedconjectures. Among these conjectures are:  - $\mathsf{DisjNP}$: The class of all disjoint NP-pairs does not havemany-one complete elements.  - $\mathsf{SAT}$: NP does not contain many-one complete sets that haveP-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 havemany-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 forthe implication $\mathsf{DisjNP}\wedge \mathsf{UP}\wedge\mathsf{NP}\cap\mathsf{coNP}\Rightarrow\mathsf{SAT}$. In particular, regardingthe conjectures by Pudl\'ak this extends a result by Khaniki [Kha19].

P-Optimal Proof Systems for Each NP-Complete Set but no Complete Disjoint NP-Pairs Relative to an Oracle