Compare up-color domination weight with Roman domination on trees

Determine whether every tree $T$ satisfies $Omega_{uc}(T)\leqgamma_r(T)$, and identify all pairs of integers $a,b\geq2$ for which some graph $G$ satisfies $Omega_{uc}(G)=a$ and $gamma_r(G)=b$.

Background

The paper compares the newly introduced up-color domination weight with the Roman domination number on complete and star graphs. It then asks whether the inequality extends to all trees and, more generally, which pairs of attainable parameter values can occur for graphs.

References

Is it true for any tree $T$ that $\Omega_{uc}(T)\leq \gamma_r(T)$?

Additionally, identify pairs of integers $a, b \geq 2$ such that there exists a graph $G$ with $\Omega_{uc}(G)=a$ y $\gamma_r(G)=b$.

— 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, sixth Problem environment and following sentence