Fujita's log spectrum conjecture on iterated accumulation points

Work over the complex numbers. For all triples $(X,\Delta,M)$, where $(X,\Delta)$ is an $n$-dimensional log smooth variety with $\Delta$ a reduced divisor and $M$ an ample Cartier divisor, let $\operatorname{PET}_n$ be the set of pseudo-effective thresholds of $M$ with respect to $(X,\Delta)$. For a subset $S\subseteq\mathbb{R}$, let $\lim^1 S$ denote the set of accumulation points of $S$, and define $\lim^k S\coloneqq\lim^1(\lim^{k-1}S)$ for $k\in\mathbb{N}$. Fujita's log spectrum conjecture. For every positive integer $k\leq n$, every element of $\lim^k(\operatorname{PET}_n)$ is less than or equal to $n-k$; equivalently,

$$\lim^k(\operatorname{PET}_n)\leq n-k.$$

The conjecture concerns the structure of pseudo-effective thresholds and is an analogue of Shokurov's conjecture on log canonical thresholds. It was settled affirmatively by the results cited in the source, so this conjecture is solved.