Kollár's restriction conjecture for local Picard groups of excellent schemes

Let $X$ be an excellent scheme that is $S_2$ and has pure dimension $\geq 4$. Let $x\in X$ be a closed point and let $x\in D\subset X$ be an effective Cartier divisor. Kollár's restriction conjecture. The kernel

$$\ker\bigl[r^X_D:\operatorname{\mathbf{Pic}}^{\rm loc}(x,X)\to\operatorname{\mathbf{Pic}}^{\rm loc}(x,D)\bigr]$$

is a unipotent algebraic group. This is a proposed extension of the local Picard-group results to excellent schemes; the source notes that the local Picard group itself is not known to exist in this generality.