The four-perfect-matchings conjecture for cubic graphs with a two-circuit 2-factor

Let $G$ be a cubic graph with a 2-factor consisting of two circuits, where a 2-factor is a spanning $2$-regular subgraph. Let the perfect matching index of $G$ be the minimum number of perfect matchings needed to cover all its edges.

Four-perfect-matchings conjecture. The perfect matching index of $G$ is at most $4$, unless $G$ is the Petersen graph.

The conjecture is presented as the paper's concluding conjecture. The preceding results establish the bound for a particular class in which the complementary $1$-factor has exactly three spokes, but the general two-circuit case remains unresolved in the supplied text.