Converse of Sun's characterization of rainbow-common graphs

Let $H$ be a graph. A graph is rainbow-common if its rainbow-uncommonness behavior does not occur for the relevant number of colors, as defined in the paper.

Converse of Sun's characterization. $H$ is rainbow-common if and only if $H$ is a forest.

This conjecture is proposed as the converse of Sun's result that every graph containing a cycle is $r$-rainbow-uncommon for all sufficiently large $r$. The paper notes that disjoint unions of stars are the only class of rainbow-common graphs currently known and suggests that the conjecture would characterize rainbow-common graphs completely.