The quasi-stable idealizer conjecture

For each $n\in\mathbb{N}$, let $X_n^\star$ be the idealizer associated with quasi-stable polynomials, and let $Y_n^\star$ be the corresponding family defined by $Y_n^\star=Y_n$ for odd $n$ and, for $n=2l$, by adjoining to $Y_{2l}$ the polynomials $g(x)=g_e(x^2)$ whose coefficient polynomial satisfies the stated finite multiplier-sequence conditions. The quasi-stable idealizer conjecture. For any $n\in\mathbb{N}$, equality $Y_n^\star=X_n^\star$ holds. The construction accounts for the additional even-degree polynomials arising from polynomials in $x^2$; the general equality is proposed beyond the cases treated in the paper.