Conjecture on control of higher eigenvalues of anticanonical powers

Let $M$ be a Fano manifold of dimension $n$. For a natural number $m$ such that $K_M^{-m}$ is very ample, consider the eigenvalues associated with $K_M^{-m}$.

Higher-eigenvalue control conjecture. For each natural number $k$ such that $2\leq k\leq n$, the $k$-th eigenvalue of $K_M^{-m}$ is controlled.

Control of these eigenvalues is the hypothesis underlying the paper's higher-alpha-invariant criterion for the existence of Kähler–Einstein metrics. The statement generalizes the control of the second eigenvalue that is essentially established in Tian's work for $k=2$; its validity for all $k$ is left as a conjecture.