Domination conjecture for neighborhoods in vertex-transitive reflection graphs

Let $H$ be a graph and $x$ a vertex. Define $H_x(k)$ to be the subgraph induced by all vertices at distance at most $k$ from $x$; for vertex-transitive $H$, write this graph as $H(k)$. Say that a graph $H$ dominates a graph $H'$ when

$$t_H(W)^{1/e(H)}\geq t_{H'}(W)^{1/e(H')}$$

for every graphon $W$.

Neighborhood domination conjecture. If $H$ is a vertex-transitive reflection graph with diameter $d$, then $H(\ell)$ dominates $H(k)$ for all $1\leq k\leq \ell\leq d$.

The conjecture proposes a monotonicity principle for domination inequalities among metric neighborhoods in vertex-transitive reflection graphs. The supplied source gives no evidence of a resolution.