Funano and Shioya's dimension-free eigenvalue ratio conjecture for Alexandrov spaces

Let $X$ be a compact finite-dimensional Alexandrov space of nonnegative curvature, and let $\lambda_k(X)$ denote the $k$-th non-zero eigenvalue of the corresponding Laplacian on $X$.

Funano and Shioya's conjecture. For every natural number $k$, there exists a positive constant $C_k$ depending only on $k$ such that

$$\lambda_k(X)\leq C_k\lambda_1(X).$$

This conjecture seeks a dimension-free bound for ratios of Laplacian eigenvalues on Alexandrov spaces, extending the corresponding estimate known for closed weighted Riemannian manifolds. The source does not provide evidence of a resolution.