The hierarchy conjecture for polynomials preserving nonnegative matrices

For $n\geq 3$, let $\mathscr{P}_n$ denote the set of polynomials that preserve entrywise nonnegative $n\times n$ matrices under polynomial evaluation. Hierarchy conjecture.

$$\mathscr{P}_{n+1}\subset \mathscr{P}_n.$$

The paper has established the inclusions $\mathscr{P}_2\subset \mathscr{P}_1$ and, by Clark and Paparella, $\mathscr{P}_3\subset \mathscr{P}_2$; the conjecture proposes that this descending hierarchy continues for every $n\geq 3$.