The conjecture on asymptotic expansion of Bergman kernels

Let $X$ be a smooth projective variety of dimension $n$, let $(L,h_L)$ be a pseudoeffective singular Hermitian line bundle on $X$, and let $A$ be a sufficiently ample line bundle on $X$ with a $C^{\infty}$ Hermitian metric $h_A$ of strictly positive curvature. Let $d\mu^{+}(L,h_L)$ and $d\mu^{-}(L,h_L)$ denote the upper and lower local volumes defined by the corresponding limsup and liminf of the normalized Bergman kernels.

Bergman-kernel asymptotics conjecture. The upper and lower local volumes coincide:

$$d\mu^{+}(L,h_L)=d\mu^{-}(L,h_L).$$

In particular, the normalized Bergman-kernel limit exists on $X$, and it equals the local volume $d\mu(L,h_L)$:

$$n!\cdot\lim_{m\rightarrow\infty}m^{-n}\cdot h_A\cdot h_L^m\cdot K(X,K_X+A+mL,h_A\cdot h_L^m)=d\mu(L,h_L).$$

The claim concerns the existence and identification of pointwise asymptotics for Bergman kernels associated with pseudoeffective singular metrics. The source presents it as a general conjecture; no resolution is supplied there.