The smallest cyclically 4-connected snark with oddness 4

A snark is a connected cubic graph with no proper 3-edge-colouring. A snark is cyclically 4-connected if no edge-cut of size at most $3$ separates two subgraphs each containing a cycle. The oddness of a snark is the minimum number of odd circuits in a 2-factor. Smallest-order conjecture. The smallest cyclically $4$-connected snark with oddness $4$ has $44$ vertices.

The construction described gives a cyclically $4$-connected snark on $44$ vertices with oddness $4$, improving the previously known smallest examples. The authors report that their other approaches also produced snarks of order $44$, but no proof of minimality is given, so the conjecture remains open.