Problem 25.23.60C91C5C2794A

Problem 25.23.6.0C91C5C2794A Consider a compact Kähler manifold $X$ and a connected submanifold $Y$. We have defined the trace operator $\operatorname{Tr}_ Y$ from a subset of $\mathrm{QPSH}(X)/\sim _\mathcal {I}$ to $\mathrm{QPSH}(Y)/\sim _\mathcal {I}$. Is it possible to refine this operator to one from a subset of $\mathrm{QPSH}(X)/\sim _ P$ to $\mathrm{QPSH}(Y)/\sim _ P$?