Demailly–Kollár openness conjecture for jumping numbers

Let $\varphi$ be a plurisubharmonic germ at a point $x$ on a complex manifold, and let $\mathfrak q\subseteq\mathcal O_x$ be a nonzero ideal. Define

$$c_x^{\mathfrak q}(\varphi)=\sup\{c>0\mid |\mathfrak q|^2\exp(-2c\varphi)\text{ is locally integrable at }x\}.$$

Demailly–Kollár jumping-number openness conjecture. If $c_x^{\mathfrak q}(\varphi)<\infty$, then $|\mathfrak q|^2\exp(-2c_x^{\mathfrak q}(\varphi)\varphi)$ is not locally integrable at $x$.

The source describes this as a generalization of the original openness conjecture and as a semicontinuity statement for multiplier ideals. Its status is open in the generality considered.