Mumford-goodness conjecture for higher direct-image metrics

Let $\mu:Y\to X$ be the birational modification in a semistable reduction, let $f:Y\to T$ be the induced family, let $F^{*}\xi$ be the pulled-back coefficient bundle, and let $h_{L^{2}}$ be the $L^{2}$-metric on the higher direct image $R^{q}f_{*}K_{Y/T}(F^{*}\xi)$. A Hermitian metric is good in the sense of Mumford if it has Mumford’s prescribed growth behavior near the boundary. Mumford-goodness conjecture. For any $q>0$, the Hermitian metric $\mu^{*}h_{L^{2}}$ on $R^{q}f_{*}K_{Y/T}(F^{*}\xi)$ should be good in the sense of Mumford. The proposition immediately preceding this statement proves goodness for the determinant metric and, under reducedness and canonical-singularity hypotheses, for the original family; the conjecture asks for the corresponding assertion for every higher direct image after pullback.