Positive-proportion conjecture for irreducible polynomials in quadratic semigroups

Let $S=\{x^2+c_1,\dots,x^2+c_r\}$ for some distinct $c_1,\dots,c_r\in\mathbb{Q}$ and $r\geq2$. Let $M_S$ denote the semigroup generated by $S$ under composition, and call a subset of $M_S$ a positive-proportion set if its proportion among elements of bounded degree has positive lower limit. Positive-proportion conjecture. $M_S$ contains a positive proportion of irreducible polynomials if and only if it contains at least one irreducible polynomial. This extends the established integer-coefficient result to rational parameters; the rational case is presented as an open problem, and the source explicitly notes that this case remains open.