The integrality conjecture for pairs of PCF unicritical parameters

Let $k$ be a number field, let $S$ be a finite set of places of $k$ including all the archimedean places, and let $d\geq 2$ be an integer. Write $f_{d,c}(z):=z^d+c$. Let $D$ be a nonzero effective divisor on $\mathbb{A}^2$ with at least one irreducible component not of any of these forms: $\{c\}\times\mathbb{A}^1$ for a PCF parameter $c$, $\mathbb{A}^1\times\{c\}$ for a PCF parameter $c$, or the solution set of $x-\zeta y=0$ for a $(d-1)$st root of unity $\zeta$. The pair-integrality conjecture. The set of points $P=(a,b)\in\mathbb{A}^2(\overline{k})$ that are $S$-integral relative to $D$ and for which both $f_{d,a}$ and $f_{d,b}$ are PCF is not Zariski dense in $\mathbb{A}^2$. This is a two-parameter integrality analogue of the Dynamical André–Oort conjecture; the paper expects density in the complementary case where every component of $D$ has one of the three exceptional forms.