Bipartite DRR classification conjecture for finite groups

Let $R$ be a finite group with a subgroup $M$ of index $2$. A bipartite DRR is a Cayley graph $\operatorname{Cay}(R,S)$, with $S\subseteq R\setminus M$, whose automorphism group is the regular representation of $R$. Bipartite DRR classification conjecture. One of the following holds: $(R,M)$ admits a bipartite DRR, or $R$ is one of the $22$ groups listed in Table 1. Thus, apart from the stated exceptional groups, every finite group with an index-$2$ subgroup admits a subset $S\subseteq R\setminus M$ producing a bipartite DRR.