Local volume versus F-signature conjecture

Let $x\in X$ be an $n$-dimensional klt singularity in characteristic $0$, and for primes $p\gg0$ let $x_p\in X_p$ be its reduction modulo $p$. Let $s(x_p,X_p)$ denote the F-signature. Local volume versus F-signature conjecture.

$$\liminf_{p\to\infty}s(x_p,X_p)\geq\frac{\widehat{\rm vol}(x,X)}{n^n}.$$

The right-hand side is the volume density, and equality holds for a smooth point; the conjecture would imply uniform positive lower bounds for these F-signatures, while the source leaves it open.