Balakrishnan et al.'s characterization conjecture for strongly distance-balanced graphs

Let $\Gamma$ be a graph. For vertices $u,v\in V(\Gamma)$, let $W_{u,v}$ denote the set of vertices closer to $u$ than to $v$, and define $d(u,S)=\sum_{x\in S}d(u,x)$ for a vertex $u$ and a subset $S\subseteq V(\Gamma)$. Balakrishnan et al.'s conjecture. A graph $\Gamma$ is strongly distance-balanced if and only if

$$d(u,W_{u,v})=d(v,W_{v,u})$$

for every pair of adjacent vertices $u,v$ of $\Gamma$. The conjecture proposes a local distance-sum characterization of strongly distance-balanced graphs, analogous to the known characterization of distance-balanced graphs; the source does not provide evidence of resolution.