K4-free signed Gyárfás–Sumner conjecture for linear forests

Let $F$ be a linear forest, meaning a forest whose connected components are paths. Let $\widehat{(K_4,-)}$ be the specified signed complete graph, and call a finite set of signed graphs a GS set when the corresponding forbidden induced-subgraph class has bounded balanced chromatic number. K4-free signed Gyárfás–Sumner conjecture. For every linear forest $F$, the set

$$\{\widehat{(K_4,-)},F\}$$

is a GS set.

The paper proves that being a linear forest is necessary and proves the claim when every component of $F$ is a path of length at most $4$. Sufficiency for arbitrary linear forests is left open.