The domatic game first-player difference conjecture

Let $G$ be a graph. Write $\operatorname{dom_g}(G,A)$ and $\operatorname{dom_g}(G,B)$ for the domatic game numbers when Alice and Bob, respectively, move first.

First-player difference conjecture. For all graphs $G$,

$$\left|\operatorname{dom_g}(G,A)-\operatorname{dom_g}(G,B)\right|\leq 1.$$

The conjecture proposes that changing which player moves first changes the domatic game number by at most one. The surrounding discussion notes that all known examples satisfy this bound, but does not establish it in general; the status is therefore open.