Gorenstein bimodule conjecture

Let $A$ be a finite-dimensional algebra. Regard $A$ as an $A$-bimodule. A module is Gorenstein-projective if it is reflexive and satisfies the relevant positive Ext-vanishing conditions against the regular module and its dual.

Gorenstein bimodule conjecture. $A$ is self-injective if and only if $A$, regarded as an $A$-bimodule, is Gorenstein-projective.

The conjecture is known when $A$ is Iwanaga–Gorenstein and, more generally, when $A$ is left weakly Gorenstein. It remains open for arbitrary finite-dimensional algebras.