Weak Gorensteinness conjecture for Eisenstein Hecke algebras

Let $\mathfrak{h}$ and $\mathfrak{H}$ be the Eisenstein components of the cuspidal and full Hecke algebras, respectively. The ideals $I$ and $I_{\mathfrak{H}}$ are their Eisenstein ideals, with $I_{\mathfrak{H}}$ the preimage of $I$. We call $\mathfrak{h}$ weakly Gorenstein if $\mathfrak{h}_{\mathfrak{p}}$ is Gorenstein for every height-one prime $\mathfrak{p}\supset I$, and similarly for $\mathfrak{H}$ using $I_{\mathfrak{H}}$. Weak Gorensteinness conjecture. The Hecke algebras $\mathfrak{h}$ and $\mathfrak{H}$ are weakly Gorenstein. This is motivated by the fact that neither algebra is Gorenstein in general; the conjecture concerns Gorensteinness after localization at height-one primes containing the relevant Eisenstein ideal.