Spectral comparison conjecture for positively curved Euclidean weighted manifolds

Let $(\mathbb{R}^n,g,\mu)$ be a connected weighted manifold satisfying $CD(\rho,\infty)$ with $\rho>0$, and let $\lambda_k(\mathbb{R}^n,g,\mu)$ denote its ordered eigenvalues. Spectral comparison conjecture. Question 1 has a positive answer for any $(\mathbb{R}^n,g,\mu)$ satisfying $CD(\rho,\infty)$ with $\rho>0$; equivalently,

$$\forall k\geq 1\qquad \lambda_k(\mathbb{R}^n,g,\mu)\geq \lambda_k(\mathbb{R}^n,\lvert\cdot\rvert,\gamma^n_\rho).$$

The conjecture restricts the general spectral comparison question to manifolds diffeomorphic to Euclidean space, after spheres provide counterexamples in the unrestricted setting. Its resolution is not supplied in the source.