Frank number conjecture for cyclically 4-edge-connected graphs

Let $G$ be a cyclically $4$-edge-connected graph, meaning that deleting fewer than $4$ edges cannot separate $G$ into two components both containing a cycle. Let $fn(G)$ denote its Frank number. Frank number conjecture for cyclically 4-edge-connected graphs. The Petersen graph is the only cyclically $4$-edge-connected graph with

$$fn(G)>2.$$

Equivalently, every cyclically $4$-edge-connected graph other than the Petersen graph should have Frank number at most $2$. The paper reduces this conjecture to the cubic case; cubic graphs have Frank number at least $2$ because they cannot be $4$-edge-connected.