Milman's contracting transport conjecture for positively Ricci-curved spheres

Let $d$ be a positive integer, let $(\mathbb{S}^d,g)$ be a Riemannian sphere with volume measure $\operatorname{vol}_g$, and suppose that

$$\operatorname{Ric}_g\geq \rho g$$

for some $\rho>0$. Let $g^\rho_{\mathrm{can}}$ be the canonical metric on $\mathbb{S}^d$ rescaled so that

$$\operatorname{Ric}_{g^\rho_{\mathrm{can}}}=\rho g^\rho_{\mathrm{can}}.$$

Milman's contracting transport conjecture. There exists a map

$$T:(\mathbb{S}^d,g^\rho_{\mathrm{can}},\operatorname{vol}_{g^\rho_{\mathrm{can}}})\to(\mathbb{S}^d,g,\operatorname{vol}_g)$$

that pushes forward $\operatorname{vol}_{g^\rho_{\mathrm{can}}}$ onto $\operatorname{vol}_g$ up to a finite constant and contracts the corresponding metrics. This is a finite-dimensional analogue of the Gaussian contraction theorem, but the supplied text gives no resolution status for it.