Higashitani–Matsumoto conjecture on Kempe classes of almost bipartite graphs

Let $G$ be a $(k-1)$-colorable $B+E_\ell$ graph, where $B$ is bipartite, $E_\ell$ denotes a set of $\ell$ added edges, $k\ge 4$, and $\ell<\binom{k}{2}$. Let $\operatorname{Kc}(G,k)$ denote the number of equivalence classes of $k$-colorings of $G$ under Kempe swaps.

Higashitani–Matsumoto conjecture.

$$\operatorname{Kc}(G,k)=1.$$

Higashitani and Matsumoto proved the conjecture for $k=4$. It was subsequently disproved for all $k\ge 8$, so the assertion does not hold in the full stated range.