Finiteness conjecture for the subclass of critically frustrated signed graphs

Let $\mathcal{L}^*(k)$ be the family of irreducible non-decomposable critically $k$-frustrated signed graphs. For each integer $m\leq k$, define $\mathcal{S}^*(k)$ to consist of those elements $(G,\sigma)$ of $\mathcal{L}^*(k)$ such that every critically $m$-frustrated subgraph of $(G,\sigma)$ is non-decomposable.

Finiteness conjecture. For every positive integer $k$, the set $\mathcal{S}^*(k)$ is finite.

This is presented as a relaxation of the finiteness conjecture for $\mathcal{L}^*(k)$; its status is not resolved in the source.