The idealizer of the semigroup of stable polynomials

It follows from the Garloff-Wagner Theorem that the set of stable polynomials of degree $n$, denoted by ${\mathcal{H}}_n$, i.e., those whose zeros all lie in the open left complex half-plane, with the Hadamard product $\ast$, forms an abelian semigroup contained in the abelian group ${\mathbb{R}}_n^{+}$ of polynomials of degree $n$ with positive real coefficients. By the idealizer of the set ${\mathcal{H}}_n$, we refer to the largest subsemigroup of ${\mathbb{R}}_n^{+}$ in which ${\mathcal{H}}_n$ is an ideal. In this paper, we formulate a conjecture characterizing the idealizer of ${\mathcal{H}}_n$ and prove it for $n⩽5$. In addition, we show that the proposed condition is necessary for any polynomial to belong to the idealizer and establish, in a distinguished special case, a sufficient condition of a similar nature that supports the conjecture.