Finiteness conjecture for non-decomposable critically frustrated signed graphs

Let $(G,\sigma)$ be a signed graph. A signed graph is critically $k$-frustrated when its frustration index is $k$ and decreases after the removal of any edge; it is non-decomposable when it is not an edge-disjoint union of critically frustrated signed graphs and is not obtained from another critically frustrated signed graph by subdivision. Let $\mathcal{L}^*(k)$ denote the family of irreducible non-decomposable critically $k$-frustrated signed graphs.

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

The cases $k=1$ and $k=2$ are explicitly classified, but finiteness for general positive integers remains open. This conjecture is also stated informally in the introduction and formally later in the paper.