Milman's measure-preserving contraction conjecture for positively Ricci-curved spheres

Let $(M^n,g)$ be a smooth Riemannian manifold diffeomorphic to $\mathbb S^n$ and suppose that

$$\operatorname{Ric}_g\geq(n-1)g.$$

Milman's contraction conjecture. There exists a map

$$T:(\mathbb S^n,g_{\mathrm{can}})\longrightarrow(M,g)$$

such that

$$T_\#\frac{\,\mathrm d\operatorname{vol}_{g_{\mathrm{can}}}}{|\mathbb S^n|}=\frac{\,\mathrm d\operatorname{vol}_g}{\operatorname{vol}_g(M)},\qquad \operatorname{Lip}(T)\leq1.$$

This conjecture is a spherical Riemannian analogue of Caffarelli's contraction theorem. The paper proves it in dimension two using inverse mean curvature flow, while the general-dimensional statement remains open.