Relative local Fujita freeness conjecture for higher direct images

Let $f:Y\to X$ be a surjective morphism from a smooth projective variety to a smooth projective variety of dimension $n$, and suppose that $f$ is smooth over $X_0=X\setminus B$ for a normal crossing divisor $B$ on $X$. Let $L$ be a nef and big invertible sheaf on $X$, and let $x\in X$ be a point. Assume that $L^n>n^n$ and $L^dZ\ge n^d$ for every irreducible subvariety $Z$ of $X$ of dimension $d$ containing $x$. Relative local Fujita freeness conjecture. For any $q\ge0$, the natural homomorphism

$$H^0(X,R^qf_*\omega_Y\otimes L)\longrightarrow R^qf_*\omega_Y\otimes L\otimes\kappa(x)$$

is surjective. This is the stronger local relative version of the preceding conjecture; the supplied text gives no evidence that it has been resolved.