Bounded neighborhoods in K4-free, induced-path-free signed graphs

For a signed graph $\widehat{G}$ and a vertex $u$, let $N[u]$ denote the closed neighborhood of $u$, and let $\chi_b$ denote balanced chromatic number. Bounded-neighborhood conjecture. For every positive integer $k$, there exists a positive integer $b_k$ such that, whenever

$$\widehat{G}\in\operatorname{Forb}_{ind}(\{\widehat{(K_4,-)},P_k\}),$$

every vertex $u$ of $\widehat{G}$ has closed neighborhood satisfying

$$\chi_b(N[u])\leq b_k.$$

This conjecture is presented as an equivalent relaxation of the conjecture for arbitrary linear forests: the paper proves that such a neighborhood bound yields a global balanced-coloring bound. It remains open.