The gap conjecture for vertices of non-degenerate 3-way transportation polytopes

Let $p \times q \times s$ be a non-degenerate $3$-way transportation polytope specified by $2$-margin matrices $U,V,W$, and let $f_0(M(U,V,W))$ denote its number of vertices. The gap conjecture. It is impossible to have

$$(p-1)(q-1)(s-1)+1 < f_0(M(U,V,W)) < 2(p-1)(q-1)(s-1).$$

The conjecture is known to hold when $p,q,s \leq 3$, but its general status is not established here.