Fleischner's dominating circuit conjecture for cyclically 4-edge-connected snarks

A snark is a bridgeless cubic graph that is not 3-edge-colourable, and a graph is cyclically 4-edge-connected if no edge cut of size less than $4$ separates two components each containing a circuit. A dominating circuit in a graph $G$ is a circuit $C$ such that every edge of $G$ has an end-vertex on $C$.

Fleischner's dominating circuit conjecture. Every cyclically $4$-edge-connected snark has a dominating circuit.

The conjecture is one of several equivalent forms of the dominating-cycle problem and is related to the Matthews–Sumner conjecture on hamiltonian claw-free graphs. The source reports computational verification that it has no counterexample on $36$ or fewer vertices, but its general status remains open.