An oracle exists relative to which TAUT has neither optimal proof systems nor recursive jump operators (even with infinite PH), showing Khaniki's question is not relativizably provable.
On Slicewise Monotone Parameterized Problems and Optimal Proof Systems for TAUT
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Recursive Jump Operators and Optimal Proof Systems
An oracle exists relative to which TAUT has neither optimal proof systems nor recursive jump operators (even with infinite PH), showing Khaniki's question is not relativizably provable.