Dong's vertex-isolation conjecture for mean color numbers

Let $G$ be a graph and let $w$ be a vertex of $G$. Write $G-w$ for the graph obtained by deleting $w$, and let $K_1$ be the one-vertex graph. Dong's vertex-isolation conjecture. One has

$$\mu(G)\geq\mu((G-w)\cup K_1).$$

This conjecture concerns whether isolating a vertex cannot increase the mean color number. Dong proved the related Bartels–Welsh lower-bound conjecture, but the supplied text gives no resolution status for this assertion.