All algebras are assumed to be connected over a field. Given an algebra $A$ with minimal faithful projective-injective left module $Af$ for some idempotent f. Let $M$ be an $A$-module of infinite dominant dimension. Then the Hochschild cohomology ring of $M$ over A should be isomorphic to the Hochschild cohomology ring of $Mf$ as $fAf$ module. Now assume we have an algebra $A$ of infinite dominant dimension, then this would give that the Hochschild cohomology rings of $A$ and $fAf$ are isomorphic. This motivates the following question:

Given an algebra $A$ with minimal faithful projective-injective left module $Af$ (of course such that A and fAf are not Morita equivalent), can $A$ and $fAf$ have isomorphic Hochschild cohomology rings?

Of course, special cases where the answer is "no" are of interest, since this yields proofs of the Nakayama conjecture in this cases. (At least when I made no mistake before)

Maybe the other direction is also true: Given an algebra $A$ with minimal faithful projective-injective left module Af such that A and fAf have isomorphic Hochschild cohomology rings, then A has infinite dominant dimension.

If it helps we can assume that B=fAf is selfinjective and A=End_B(M) for some generator-cogenerator M of B.