Structure of minimum cycle-separating cuts in cages

Prove that, for every $(k,g)$-cage, every cycle-separating $(k-2)g$-edge-cut separates a cycle of length $g$.

Background

The second conjecture strengthens the proposed maximal cyclic edge-connectivity statement by specifying the structure of every cut of the conjectured minimum possible size. It asserts that such a cut must isolate a shortest cycle, namely a cycle whose length equals the girth. The paper establishes this strengthening for cages of order at most $2M(k,g)-g^2$.