Contraction conjecture for positively curved Euclidean weighted manifolds

Let $(\mathbb{R}^n,g,\mu)$ satisfy $CD(\rho,\infty)$ with $\rho>0$, and let $\gamma^n_\rho$ be the $n$-dimensional Gaussian measure. A map pushes forward $\gamma^n_\rho$ onto $\mu$ up to a finite constant if its pushforward measure is proportional to $\mu$, and it contracts the corresponding metrics when it is $1$-Lipschitz. Contraction conjecture. There exists a map

$$T:(\mathbb{R}^n,\lvert\cdot\rvert,\gamma^n_\rho)\longrightarrow(\mathbb{R}^n,g,\mu)$$

pushing forward $\gamma^n_\rho$ onto $\mu$ up to a finite constant and contracting the corresponding metrics. Motivated by Caffarelli’s contraction theorem and Gaussian comparison results, this conjecture would imply the spectral comparison conjecture through the contraction principle. The source gives no resolution.