Folklore eventual monotonicity conjecture for alpha-invariants

Let $X$ be a projective manifold and let $L$ be an ample line bundle on $X$. For each positive integer $k$, let $\alpha_k(L)$ denote the level $k$ log canonical threshold. The sequence $(\alpha_k(L))_{k\geq1}$ is the sequence of quantized alpha-invariants.

Folklore monotonicity conjecture. There exists $k_0\in\mathbb{N}_+$ such that, whenever $k_0\leq k_1\leq k_2$,

$$\alpha_{k_1}(L)\geq\alpha_{k_2}(L).$$

This weaker folklore conjecture predicts that the quantized alpha-invariants are eventually monotone. The paper states that it also provides a counterexample to this conjecture, so it is refuted.