Asymptotic constant conjecture for vertices of polystochastic matrix polytopes

For integers $d,n\geq 2$, let $\Omega_n^d$ be the polytope of $d$-dimensional polystochastic matrices, and let $V(n,d)$ denote its number of vertices. For fixed $d$, write $n\to\infty$. Asymptotic constant conjecture. For every $d\geq 2$, there is a constant $c_d$, $1\leq c_d\leq d$, such that

$$\ln V(n,d)=c_d n^{d-1}\ln n\cdot(1+o(1)).$$

The preceding bounds establish the claimed scale and give the exact value $c_2=1$, while for $d=3$ they place $c_3$ between $3/2$ and $3$, and for fixed $d\geq 4$ between $1$ and $d$. Determining whether such constants exist, and their values in higher dimensions, remains open.