Equivalent local boundedness conjecture for signed graph classes

Let $(K_3,-)$ denote the signed triangle with the specified signature, let $(K_4,M)$ denote the signed complete graph whose negative edges form a matching of size two, and let $P_k$ be a path on the conventionally specified number of vertices. Equivalent local boundedness conjecture. For every positive integer $k$, there exists a positive integer $b_k$ such that every signed graph in

$$\operatorname{Forb}_{ind}(\{(K_3,-),(K_4,M),P_k\})$$

has balanced chromatic number at most $b_k$.

The paper gives this as a reformulation of the bounded-neighborhood conjecture, using switching and the structure of the neighborhood of a vertex. It remains open.