Milman's Gaussian spectral conjecture under the CD condition

Let $d$ be a positive integer, let $(\mathbb{R}^d,g,\mu)$ satisfy the curvature-dimension condition $\operatorname{CD}(\rho,\infty)$ for some $\rho>0$, and let $\lambda_k(\mathbb{R}^d,g,\mu)$ denote the $k$th eigenvalue in the associated weighted spectral problem. Define the centered Gaussian probability measure by

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

where $c_\rho^d>0$ is a normalization constant. Milman's Gaussian spectral conjecture. For every $k\geq 1$,

$$\lambda_k(\mathbb{R}^d,g,\mu)\geq\lambda_k(\mathbb{R}^d,|\cdot|,\gamma_\rho^d).$$

This is the infinite-dimensional-model analogue in Euclidean space of the spherical spectral comparison conjecture. The supplied text gives no evidence that this claim has been resolved.