Resistance-distance bound for connected balanced digraphs

Let $D$ be a finite connected balanced digraph. For vertices $i,j$ of $D$, let $r_{ij}^{D}$ denote their resistance distance, and let $d_{ij}^{D}$ denote the length of a shortest directed path from $i$ to $j$ in $D$.

Resistance-distance conjecture. The resistance distance is bounded above by the directed distance:

$$r_{ij}^{D} \leq d_{ij}^{D}.$$

The paper notes that connected balanced digraphs are strongly connected, making this equivalent to the corresponding statement for strongly connected balanced digraphs. The conjecture was proved using combinatorial methods by counting spanning forests satisfying certain conditions.