Daskalopoulos' symmetry conjecture for ancient ovals

Let an ancient oval be an ancient compact noncollapsed mean curvature flow that is not self-similar, and let $1\leq k\leq n$. An ancient oval is $\mathrm{O}(k)\times \mathrm{O}(n+1-k)$-symmetric if it is invariant under this product of orthogonal groups. Daskalopoulos' conjecture. Every ancient oval is $\mathrm{O}(k)\times \mathrm{O}(n+1-k)$-symmetric for some $k$. Ancient ovals arise as potential singularity models for mean-convex flows and are relevant to the fine structure of singularities. The conjecture asks whether all ancient ovals possess one of the orthogonal symmetries occurring in the known constructions; its resolution is not supplied here.