How strong have the oracles needed to be for these two classes to be proven equivalent with respect to them?

For instance: is $ P^H$ = $ NP^H$ (ie. is $ P$ equipped with an oracle to solve the halting problem equivalent to $ NP$ equipped with an oracle to solve the halting problem)?

From Theodore Baker, John Gill, and Robert Solovay. Relativization of the P=?NP problem. Siam Journal of Computing, 4:432-442, 1975 [219] we know $ NP^A =P^A$ for their oracle A (which is a decision algorithm for a PSPACE complete problem).

If the oracle can perform an infinite amount of computation and return back the result in one step are these classes equal with respect to an oracle of this type? How about weaker ones? What is the weakest oracle we know of where $ P$ and $ NP$ are equal with respect to it?

An answer I’m looking for is something like: $ P^O$ =$ NP^O$ with respect to an oracle O and any oracle more powerful than it.