The finite quadratic-form characterization conjecture for matrix-preserving polynomials

Let $\mathcal{P}_{n,2n}$ be the cone of polynomials of degree at most $2n$ that preserve nonnegative matrices of order $n$, and let $\mathbf{v}$ denote the coefficient vector of such a polynomial. For a polynomial $f_k\in\mathbb{R}^{2n}[x]$, write $f_k(\mathbf{v})$ for its associated quadratic-form condition. Finite quadratic-form characterization conjecture. There exists a finite $k\in\mathbb{Z}^+$ such that $f_k(\mathbf{v})\geq 0$ for $f_k\in\mathbb{R}^{2n}[x]$ implies that the polynomial whose coefficients are $\mathbf{v}$ belongs to $\mathcal{P}_{n,2n}$. Equivalently, finitely many quadratic forms should describe the boundary of $\mathcal{P}_{n,2n}$; the conjecture is proposed as a route toward characterizing this cone.