Petruševski–Škrekovski edge-deletion conjecture for odd edge-colorings

Let $G$ be a connected graph that is not a Shannon triangle of type $(2,2,2)$ or type $(2,2,1)$. Write $\chi_{\text{odd}}'(G)$ for the odd chromatic index, and call an edge special if its removal makes the graph odd $3$-edge-colorable. Petruševski–Škrekovski edge-deletion conjecture. Either

$$\chi_{\text{odd}}'(G)\leq 3,$$

or there exists a special edge $e$ whose removal makes the graph odd $3$-edge-colorable. Equivalently, assuming $\chi_{\text{odd}}'(G)=4$, there exists an odd $4$-edge-coloring in which color $4$ is used only once. The paper states that this conjecture is resolved by its results; it is the edge analogue of the vertex-deletion part of the odd edge-coloring theory.