Reduce the color bound for optimal up-color domination weight colorings

Determine whether every graph $G$ has an $Omega_{uc}$-domination coloring $c$ attaining $Omega_{uc}(G)$ while using fewer than $2chi(G)$ colors.

Background

Theorem 4.1 constructs an optimal up-color domination coloring using at most $2chi(G)-1$ colors. The authors observe that their proof does not use more than this number, but explicitly ask whether a strictly smaller bound, expressed as fewer than $2chi(G)$ colors, always exists.

References

Actually, in the proof of Theorem~\ref{th:Omegachigamma} we never use more than $2\chi(G)-1$ colors. Is it true that there exists always an $\Omega_{uc}$--domination coloring $c$ of $G$ ($\omega_{uc}(G,c)=\Omega_{uc}(G)$) using fewer than $2\chi(G)$ colors?

— Domination on Vertex-weighted Graphs Induce by a Coloring  (2502.07248 - Garrido-Vizuete et al., 11 Feb 2025) in Section 5, Conclusions and open problems, fourth Problem environment