Equality of parity edge-coloring numbers for bipartite graphs

Let $G$ be a bipartite graph. Let $p(G)$ and $\widehat{p}(G)$ denote the minimum numbers of colors in a parity edge-coloring and a strong parity edge-coloring, respectively. The bipartite equality conjecture.

$$p(G)=\widehat{p}(G).$$

The paper gives examples with $\widehat{p}(G)>p(G)$, but all such examples contain odd cycles; whether equality holds for every bipartite graph remains open.