Quantized volume comparison conjecture for K-semistable Fano manifolds

Let $X$ be a K-semistable Fano manifold of dimension $n$, and let $m$ be a positive integer. The anticanonical Hilbert function of $X$ is compared with that of projective $n$-space.

Quantized volume comparison conjecture. For every $m\geq 1$,

$$\dim H^0(X,-mK_X)\leq \dim H^0({\mathbb P}^n,-mK_{{\mathbb P}^n}).$$

If equality holds for some $m$, then $X\cong {\mathbb P}^n$.

The preceding theorem establishes the same inequality, and the same rigidity statement, for all $m\geq m_0$, where $m_0$ depends only on $n$. The conjecture asks for the bound uniformly for every positive integer $m$.