Minimal signed Gyárfás–Sumner sets containing a negative edge

Let a finite set of signed graphs be a GS set when its forbidden induced-subgraph class has bounded balanced chromatic number, and call it minimal if no proper subset is a GS set. Minimal-set conjecture. Any minimal finite GS set containing $\widehat{(K_2,-)}$ has exactly three elements.

This statement is the signed formulation of the classical Gyárfás–Sumner conjecture for the hereditary class corresponding to graphs with no negative simple edge. The paper presents it as an open conjectural description of minimal finite GS sets.