 Solution

Root the graph at $x_0$ and let $P^{(0)}$ be the transition matrix of simple random walk killed on hitting $x_0$, indexed by the remaining vertices. Its Green matrix is $G=(I-P^{(0)}{)}^{-1}$. Successively eliminating vertices in an order $x_1,\dots ,x_n$ takes Schur complements; the corresponding diagonal pivot at step $j$ is exactly $g_{D\setminus \{x_0,\dots ,x_{j-1}\}}(x_j)$. The product of the pivots is thereforewhich is invariant under the elimination order.