Boundedness up to special degeneration from positive local volume

Let $n$ be a positive integer, and let $\epsilon,\eta$ be positive numbers. A special degeneration of a klt singularity is the central fiber of a special test configuration as defined in the paper; a set of singularities is log bounded up to special degeneration if it admits the corresponding bounded model after such a degeneration. Let $x\in(X,\Delta)$ be an $n$-dimensional klt singularity, and let $\widehat{\operatorname{vol}}(x,X,\Delta)$ denote its local volume. Boundedness conjecture up to special degeneration. The set of such singularities satisfying

$$\widehat{\operatorname{vol}}(x,X,\Delta)\geq\epsilon$$

and having all coefficients of $\Delta$ at least $\eta$ is log bounded up to special degeneration.

This conjecture would extend the known boundedness theorem for $\mathbb{Q}$-Gorenstein singularities admitting $\delta$-plt blow-ups. The paper proves the claim in several cases, but the general statement remains open.