Compactness of increasing nets of non-Archimedean psh metrics

Let $X$ be a unibranch variety, let $L$ be a line bundle on $X$, and consider the space of non-Archimedean plurisubharmonic metrics $\mathrm{PSH}(L^{\mathrm{an}})$. An increasing net in this space is bounded from above if there is an upper bound for all its elements. Compactness conjecture for increasing nets. Every bounded from above increasing net of elements in $\mathrm{PSH}(L^{\mathrm{an}})$ converges in $\mathrm{PSH}(L^{\mathrm{an}})$. The space of non-Archimedean psh metrics has many expected properties of psh functions, but this compactness result is stated as not known in the source.