= 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 $x_1,\ldots,x_n$ takes <Schur complements>; the corresponding diagonal pivot at step $j$ is exactly $g_{D\setminus\{x_0,\ldots,x_{j-1}\}}(x_j)$. The product of the pivots is therefore
$$
\det G=\frac1{\det(I-P^{(0)})},
$$
which is invariant under the elimination order.
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