Fujita's log spectrum conjecture for pseudo-effective thresholds

Let $(X,\Delta)$ be a log smooth variety with $\Delta$ a reduced divisor. Let $M$ be an ample Cartier divisor on $X$, and let $\operatorname{pet}(X,\Delta;M)\in\mathbb{R}_{\geq 0}$ be the pseudo-effective threshold of $K_X+\Delta$ with respect to $M$. Let $\operatorname{PET}_n$ be the set of all such thresholds for varieties of dimension $n$. For $S\subseteq\mathbb{R}$, let $\lim^1S$ be the set of accumulation points of $S$, and define

$$\lim^kS=\lim^1(\lim^{k-1}S)$$

for $k\in\mathbb{N}$.

Fujita's log spectrum conjecture. Under this notation,

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

for any positive integer $k\leq n$.

The conjecture is an analogue for pseudo-effective thresholds of the conjecture on accumulation points of log canonical thresholds. The source notes that the related ACC conjecture for pseudo-effective thresholds has been proved in several settings, but gives no resolution status for this log spectrum conjecture.