Ban and Linial's finiteness conjecture for regular graphs without internal partitions

Let $d$-regular graphs be graphs in which every vertex has degree $d$, and call a partition of the vertex set into two parts an internal partition when every vertex has at least as many neighbors in its own part as in the other part. Ban and Linial's conjecture. For every $d$ there are only finitely many $d$-regular graphs with no internal partitions. The conjecture proposes that graphs lacking internal partitions are exceptional within each fixed regularity degree; the source gives no resolution here.