The idealizer conjecture for stable polynomials

For each $n\in\mathbb{N}$, let $X_n\subset\mathbb{R}_n^+$ be the largest subsemigroup containing $\mathcal{H}_n$ and $\mathbb{I}_n$ such that, for every $m\leq n$, Hadamard multiplication by an element of $X_n$ maps $\mathcal{H}_m$ into $\mathcal{H}_m$. Let $Y_n$ be the family of polynomials defined in the paper, with $Y_1=X_1$, $Y_2=X_2$, $Y_3=X_3$, and $Y_4=X_4$. The idealizer conjecture. For any $n\in\mathbb{N}$, equality $Y_n=X_n$ holds. The cases $n\leq 4$ are established in the paper, while the conjecture proposes the equality for all degrees.