Long-time diffusion mean conjecture for compact symmetric spaces

Let $M$ be a connected compact symmetric space with a smooth isometric embedding into a Euclidean space that is unique up to Euclidean isometries. Let the diffusion mean set be the set of diffusion means at diffusion time $t$, and let the extrinsic mean set be the mean set defined using the chordal distance induced by the embedding. Long-time diffusion mean conjecture. As $t\to\infty$, the diffusion mean set converges to the extrinsic mean set in this isometric embedding in the sense of the upper and lower Kuratowski limits. The conjecture extends the known long-time result for circles and spheres and is motivated here by the corresponding result for real projective spaces; it remains open for general connected compact symmetric spaces.