Tian's stabilization conjecture for alpha-invariants

Let $X$ be a projective manifold and let $L$ be an ample line bundle on $X$. The global log canonical threshold is denoted by $1\alpha(L)$, and the level $k$ log canonical threshold by $1\alpha_k(L)$. By the known relation $\alpha=\inf_k\alpha_k=\lim_{k\to\infty}\alpha_k$, these are the alpha-invariant and its quantized versions.

Tian's stabilization conjecture. There exists $k_0\in\mathbb{N}_+$ such that, for every $k\geq k_0$,

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

The conjecture asserts eventual stabilization of the quantized alpha-invariants and is motivated by their connection with Kähler–Einstein metrics. The paper provides a counterexample, so this conjecture is refuted.