Akiyama–Pethő integrality conjecture for contractive-polynomial volumes

Let $\mathcal{E}_d^{(s)}$ be the set of coefficient vectors of monic degree-$d$ real polynomials whose roots include exactly $s$ pairs of nonreal conjugates, and let

$$v_d^{(s)}=\lambda_d(\mathcal{E}_d^{(s)}),\qquad v_d^{(0)}=\lambda_d(\mathcal{E}_d^{(0)}).$$

Here $d$ is the polynomial degree and $s\leq d/2$. Akiyama–Pethő integrality conjecture. The quotient

$$\frac{v_d^{(s)}}{v_d^{(0)}}$$

is an integer for all $s\leq d/2$. The volumes are rational, and the quotient is known to be an odd integer when $s=0$; integrality in the general case was proposed on the basis of numerical experiments.