The maximal-singular-locus conjecture for K-orbit closures

Let $Y_{\gamma}$ be a $K$-orbit closure and let $\mathcal O_{\alpha}$ be a $K$-orbit appearing in the maximal singular locus of $Y_{\gamma}$. Maximal-singular-locus conjecture. All arcs of $\alpha$ match adjacent vertices, and the number of arcs of $\alpha$, equivalently the dimension of $\mathcal O_{\alpha}$, is weakly bounded above by the number of arcs of $\gamma$. The paper presents this as a proposed description of maximal singular strata; no general proof is given.