The edge-contraction lower-bound conjecture for dominated chromatic number

Let $G$ be a graph and let $e$ be an edge of $G$. Write $G/e$ for the graph obtained by contracting $e$, and let $\chi_{dom}(G)$ denote the dominated chromatic number of $G$. Edge-contraction conjecture.

$$\chi_{dom}(G)-1\leq \chi_{dom}(G/e).$$

The authors report that this stronger lower bound is suggested by checking graphs of small order, while they have not been able to prove it. It improves the previously established bound $\chi_{dom}(G)-2\leq \chi_{dom}(G/e)$ by one.