Large-girth strong arboricity conjecture

Let $G$ be a graph. Its arboricity $\operatorname{arb}(G)$ is the least number of colors in an edge coloring with no monochromatic cycle, and its strong arboricity $\zeta(G)$ is the least number of colors in a coloring that remains acyclic after contracting any single edge. The girth of $G$ is the length of its shortest cycle.

Large-girth strong arboricity conjecture. For every integer $k$ there is an integer $g(k)$ such that every graph $G$ with

$$\operatorname{arb}(G)\leqslant k$$

and girth at least $g(k)$ satisfies

$$\zeta(G)=\operatorname{arb}(G).$$

The conjecture expresses the expectation that sufficiently large girth makes strong arboricity attain its smallest possible value, namely arboricity, within every bounded-arboricity class.