Morton–Vivaldi irreducibility conjecture for quadratic delta factors

Let $f_c(z)=z^2+c$. For positive integers $m,n$ with $m\mid n$, define the delta factors $\Delta_{n,m}$ by

$$\Delta_{n,m}(c)\coloneqq \operatorname{Res}_x\bigl(\Phi^{\mathrm{cyc}}_{n/m}(x),\delta_m(x,c)\bigr) \quad (m<n)$$

and, when $m=n$, by

$$\delta_n(1,c)=\Delta_{n,n}(c)\prod_{m\mid n,\,m\neq n}\Delta_{n,m}(c).$$

Here $\Phi^{\mathrm{cyc}}_k$ is the $k$-th cyclotomic polynomial, and the roots of $\Delta_{n,m}$ are the parabolic parameters of $f_c$ associated with the corresponding periods and multipliers. Morton–Vivaldi irreducibility conjecture. For positive integers $m,n$ with $m\mid n$, every polynomial $\Delta_{n,m}$ is irreducible over $\mathbb{Q}$. Morton and Vivaldi proposed this as a number-theoretic property of parabolic parameters for the quadratic family. The conjecture is cited as an open problem, although the paper establishes irreducibility results for particular cases.