Relative minimal singularity metric extension conjecture

Let $f:X\to S$ be a surjective proper Kähler morphism with connected fibers between connected complex manifolds, let $S^{\circ}$ be the complement of the discriminant locus, and let $(L,h_L)$ be a semipositive hermitian line bundle on $X$. Suppose that $K_{X_s}+L|_{X_s}$ is pseudoeffective for every $s\in S^{\circ}$. For each $s\in S^{\circ}$, let $dV_{\max}((L,h_L)|_{X_s})$ be the maximal volume form defined as the upper semicontinuous envelope of the pointwise supremum of admissible inverse metrics, and define the relative volume form by

$$dV_{\max,X/S}(L,h_L)|_{X_s}:=dV_{\max}((L,h_L)|_{X_s}).$$

Define the metric on $(K_{X/S}+L)|_{f^{-1}(S^{\circ})}$ by taking the lower semicontinuous envelope of $dV_{\max,X/S}(L,h_L)^{-1}\cdot h_L$. Relative minimal singularity metric extension conjecture. The metric $h_{\min,X/S}(L,h_L)$ extends to a singular hermitian metric on $K_{X/S}+L$ over $X$ and has semipositive curvature. This would extend the fiberwise construction of canonical metrics to relative Kähler fibrations; the supplied text gives no resolution status.