Facet-count conjecture for three-orbit cyclic permutation polytopes

Let $a,b,c\geq 2$ be pairwise coprime integers, and let $P(a,b,c)$ be the permutation polytope associated with three disjoint cycles of lengths $ab$, $ac$, and $bc$. Facet-count conjecture. If $b,c\geq 3$ are odd and coprime, then the number of facets of $P(2,b,c)$ is

$$(2^b-2)(2^c-2)+2b+2c+bc.$$

This conjecture is motivated by a lower bound proved immediately beforehand and by computational facet counts; the authors note that the lower bound appears optimal when $a=2$, but the conjecture remains unproved in the supplied text.