Bounded nonrepetitive connection conjecture for 2-connected graphs

Let $G$ be a graph, and let $\operatorname{nrc}(G)$ denote the least number of colors in a nonrepetitive connected coloring of $G$.

Bounded nonrepetitive connection conjecture. There exists a constant $t$ such that every $2$-connected graph $G$ satisfies

$$\operatorname{nrc}(G)\leq t.$$

The paper establishes finite bounds for several more highly connected classes, including $4$-connected graphs, but leaves the existence of a uniform bound for all $2$-connected graphs open.