3-connected minimum-degree conjecture for odd edge-colorings

Let $G$ be a $3$-connected graph of odd order, and let $\delta(G)$ denote its minimum degree.

3-connected minimum-degree conjecture. If

$$\delta(G)\geq 4,$$

then

$$\chi'_o(G)\leq 3.$$

The conjecture is motivated by the proved result for $4$-connected graphs of odd order and by constructions of $3$-connected graphs with odd edge-chromatic number $4$ that contain vertices of degree $3$.