Dong's one-edge-at-a-vertex conjecture for mean color numbers

Let $G$ be a graph, let $w$ be a vertex of $G$ with $d(w)\geq 1$, and let $H$ be obtained from $G$ by deleting all but one of the edges incident to $w$. Dong's one-edge-at-a-vertex conjecture. One has

$$\mu(G)\geq\mu(H).$$

This is a proposed monotonicity principle for the mean color number under deletion of edges incident to one vertex. The supplied text gives no resolution status for this conjecture.