The non-Gorenstein-locus conjecture for K-orbit closures

Let $Y_{\gamma}$ be a $K$-orbit closure, and suppose the maximal components of its singular locus are $\mathcal O_{\alpha_1},\ldots,\mathcal O_{\alpha_k}$. Non-Gorenstein-locus conjecture. The orbit $\mathcal O_{\beta}$ appears in the non-Gorenstein locus of $Y_{\gamma}$ if and only if

$$\beta\leq\alpha_i$$

for some $i$ such that $Y_{\gamma}$ is non-Gorenstein along $\mathcal O_{\alpha_i}$. This proposes that the non-Gorenstein locus is determined by the generic singularities along the maximal components of the singular locus; the assertion remains open.