Basic semialgebraicity conjecture for symmetric nonnegative realizable spectra

Let $n$ be a positive integer. A spectrum is realizable by a symmetric nonnegative matrix if it is the spectrum of an $n\times n$ symmetric matrix with nonnegative entries. A subset of a real affine space is basic semi-algebraic if it can be described by finitely many polynomial equalities and inequalities.

Basic semialgebraicity conjecture. The set of spectra realizable by symmetric nonnegative matrices forms a basic semi-algebraic set.

This is the corresponding basic-semialgebraic-set question for the symmetric nonnegative inverse eigenvalue problem. If established, the result would provide a finite description by polynomial inequalities, addressing the difficulty of controlling the number of inequalities in the semi-algebraic approach.