Uniqueness of bubble limits at a singular Einstein point

Let $(X_j,g_j,p_j)$ be a sequence of Einstein metrics converging with non-collapsing volume to a singular Einstein metric $(X_\infty,g_\infty,p_\infty)$, and let $\operatorname{Vol}(p_\infty)$ denote the volume density at $p_\infty$. A bubble limit is a pointed geometric limit obtained by rescaling around $p_\infty$. Cone rigidity conjecture. The set of bubble limits whose volume density equals $\operatorname{Vol}(p_\infty)$ consists of a single point. Uniqueness of tangent cones is known when curvature has only quadratic blow-up at $p_\infty$, but the general uniqueness assertion remains an open folklore question.