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We build on a working program initiated by Pudl\'ak [Pud17] and construct anoracle relative to which each $\mathrm{coNP}$-complete set has$\mathrm{P}$-optimal proof systems and $\mathrm{NP}\cap\mathrm{coNP}$ does nothave complete problems.

$\mathrm{P}$-Optimal Proof Systems for Each $\mathrm{coNP}$-Complete Set  and no Complete Problems in $\mathrm{NP}\cap\mathrm{coNP}$ Relative to an  Oracle