The satellite $A$-polynomial formula for integer pseudo-graph knots

Let $C\in\mathcal{G}_{\mathbb{Z}}$ have strongly detected boundary slopes $\mathcal{DS}_{C}$, and let $K=\operatorname{Sat}(P,C,f)$ be a satellite knot whose embedded pattern knot $f(P)\subset V$ has winding number zero in $V$. For each $r\in\mathcal{DS}_{C}$, let $f(P)_r$ be the knot obtained from $f(P)$ in the $(1/r)$-Dehn filling of $V$, and let $\widetilde{A}_{f(P)_r}$ have the notation of the preceding theorem. Satellite $A$-polynomial conjecture. The $A$-polynomial of $K=\operatorname{Sat}(P,C,f)$ is given by

$$A_K=\operatorname{Red}\left[(L-1)\prod_{r\in\mathcal{DS}_{C}}\widetilde{A}_{f(P)_r}\right].$$

This extends the preceding theorem from $C\in\mathcal{G}_{0}$ to $C\in\mathcal{G}_{\mathbb{Z}}$ and would provide a method for calculating the $A$-polynomials of many satellite knots; no resolution is stated.