Milman's spectral conjecture for positively Ricci-curved spheres

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

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

Here $\lambda_k(\mathbb{S}^d,g,\operatorname{vol}_g)$ denotes the $k$th eigenvalue in the relevant weighted spectral problem. Milman's spectral conjecture. For every $k\geq 1$,

$$\lambda_k(\mathbb{S}^d,g,\operatorname{vol}_g)\geq\lambda_k(\mathbb{S}^d,g^\rho_{\mathrm{can}},\operatorname{vol}_{g^\rho_{\mathrm{can}}}).$$

This is one of Milman's proposed finite-dimensional extensions of the Gaussian contraction paradigm. The paper states that its corresponding conjecture is false in general, so this spectral claim is refuted.