Milman's contracting Gaussian transport conjecture under the CD condition

Let $d$ be a positive integer, let $(\mathbb{R}^d,g,\mu)$ satisfy $\operatorname{CD}(\rho,\infty)$ for some $\rho>0$, and define the centered Gaussian probability measure

$$\gamma_\rho^d(\mathrm{d}x)=c_\rho^d e^{-\rho|x|^2/2}\,\mathrm{d}x,$$

where $c_\rho^d>0$ is a normalization constant. Milman's contracting Gaussian transport conjecture. There exists a map

$$T:(\mathbb{R}^d,|\cdot|,\gamma_\rho^d)\to(\mathbb{R}^d,g,\mu)$$

that pushes forward $\gamma_\rho^d$ onto $\mu$ up to a finite constant and contracts the corresponding metrics. This is the transport-map counterpart of the Gaussian spectral comparison conjecture. The supplied text gives no evidence that this claim has been resolved.