Bounded chromatic number for bounded-degree color classes with no lonely colors

Let $G$ be a graph whose edges are colored, and suppose that the subgraph induced by every color has maximum degree $d$. Say that a color is lonely on a cycle if it appears exactly once on that cycle.

No-lonely-color conjecture. There is a function $f$ such that, whenever no color is lonely on any cycle of $G$, one has

$$\chi(G)\leq f(d).$$

This conjecture strengthens the no-lonely-color variant of the question discussed in the paper. The supplied status evidence says that the related existence question for graphs with a one popular color coloring remains open; the conjecture itself is therefore recorded as open.