Finiteness conjecture for prime critically frustrated signed graphs

A signed graph $(G,\sigma)$ is a graph with a positive or negative sign on each edge. Its frustration index $l(G,\sigma)$ is the minimum number of negative edges among signatures switching equivalent to $\sigma$. A signed graph is critically $k$-frustrated if $l(G,\sigma)=k$ and deleting any edge decreases the frustration index. It is prime if it is irreducible and contains no pair of edge-disjoint negative cycles. Finiteness conjecture. For every positive integer $k$, there are only finitely many prime critically $k$-frustrated signed graphs. This is the finiteness question arising from the study of primitive critical signed graphs and the multiple weak $2$-linkage problem. The supplied text states that the cases of prime critically 1-frustrated and 2-frustrated signed graphs were subsequently classified, but it does not establish the conjecture for all positive integers $k$.