Quantization conjecture for the alpha invariant of compact group actions

Let $(M,L)$ be a polarized manifold and let $\hat K$ be a compact subgroup of $\operatorname{Aut}(M)$. Suppose that $L$ is $\hat K$-linearized. Let $H_{\hat K}^0(M,L^m)$ be the maximal subspace of $H^0(M,L^m)$ spanned by a basis of holomorphic sections lying in one-dimensional $\hat K$-invariant subspaces of $H^0(M,L^m)$. If, in addition, the graded ring

$$\bigoplus_{m\in\mathbb N}H_{\hat K}^0(M,L^m)$$

is finitely generated, then the quantization conjecture for the $\hat K$-alpha invariant.

$$\alpha_{m_0l,1}^{\hat K}(M,L)=\alpha^{\hat K}(M,L),\qquad \forall~l\in\mathbb N_+.$$

This is proposed as a quantization modification of Tian's conjecture for $\alpha_{m,1}^{\hat K}(M,L)$. The supplied text does not state whether it has been proved or disproved.