The normalized-volume decay conjecture for matrix-preserving polynomial cones

Let $\mathcal{P}_{n,k}$ be the cone of polynomials of degree at most $k$ that preserve nonnegative matrices of order $n$, let $\overline{B}_{k+1}(0,1)$ be the unit ball in coefficient space, and let $\mathcal{H}^{k+1}$ denote $(k+1)$-dimensional Hausdorff measure. For $k\geq 2n$, consider the normalized volume

$$\frac{\mathcal{H}^{k+1}\left(\mathcal{P}_{n,k}\cap\overline{B}_{k+1}(0,1)\right)}{\mathcal{H}^{k+1}\left(\overline{B}_{k+1}(0,1)\right)}.$$

Normalized-volume decay conjecture.

$$\lim_{k\rightarrow\infty}\frac{\mathcal{H}^{k+1}\left(\mathcal{P}_{n,k}\cap\overline{B}_{k+1}(0,1)\right)}{\mathcal{H}^{k+1}\left(\overline{B}_{k+1}(0,1)\right)}=0.$$

This is one of the paper's two conjectures about the asymptotic volume of these cones; the preceding finite-dimensional volume comparison is proved, while this limiting assertion remains open.