The 3-connected degree-swappability conjecture

Let $G$ be a graph. A degree-assignment for $G$ assigns to each vertex $v$ a list $L(v)$ of colors with $|L(v)|=d_G(v)$, and two $L$-colorings are $L$-equivalent if one can be obtained from the other by a sequence of Kempe changes. The graph $G$ is degree-swappable if, for every degree-assignment $L$, all $L$-colorings of $G$ are $L$-equivalent.

Degree-swappability conjecture. Every $3$-connected graph distinct from the complete graph and the triangular prism is degree-swappable.

This conjecture would strengthen the theorem that every $4$-connected graph distinct from the complete graph is degree-swappable. Earlier work gives examples of $2$-connected graphs that are not degree-swappable, so the conjecture concerns the boundary between these known negative and positive results.