Jaeger–Zhang conjecture for signed planar graphs

Let $p$ be a positive integer, and let $\mathcal{P}^*_{4p+1}$ denote the class of signed planar graphs with the indicated girth condition. Jaeger–Zhang conjecture.

$$\chi_c(\mathcal{P}^*_{4p+1})\leq \frac{2p+1}{p}.$$

This conjecture gives an upper bound for the circular chromatic numbers of the corresponding classes of signed planar graphs; the paper presents it as a notable conjecture, while the surrounding discussion indicates that determining the exact values and closing the known bounds remain open problems.