Tian's strong stabilization conjecture for alpha-invariants

Let $X$ be a projective manifold and let $L$ be an ample line bundle on $X$. Define the section ring

$$R(X,L):=\bigoplus_{k=0}^{\infty}H^0(X,kL).$$

The quantities $\alpha(L)$ and $\alpha_k(L)$ denote respectively the global and level $k$ log canonical thresholds.

Tian's strong stabilization conjecture. If $R(X,L)$ is generated by

$$\bigoplus_{k=0}^{k_0}H^0(X,kL),$$

then, for every $k\geq k_0$,

$$\alpha_k(L)=\alpha(L).$$

In particular, if $R(X,L)$ is generated by $H^0(X,L)$, then, for every $k\in\mathbb{N}_+$,

$$\alpha_k(L)=\alpha(L).$$

This is a stronger form of Tian's stabilization conjecture because it predicts an explicit stabilization threshold from generators of the section ring. A counterexample was provided by Ahmadinezhad, Cheltsov, and Schicho, so the conjecture is refuted.