Finite linear parametrization conjecture for equal-volume Dehn fillings

Let $\mathcal{M}$ be a $1$-cusped hyperbolic $3$-manifold, and write $\mathcal{M}_{p/q}$ for its hyperbolic Dehn filling along slope $p/q$. Finite linear parametrization conjecture. There exists a finite collection

$$\{(a_i,b_i,c_i,d_i)\in\mathbb{Z}^4\}_{1\leq i\leq k}$$

such that if

$$\operatorname{vol}(\mathcal{M}_{p/q})=\operatorname{vol}(\mathcal{M}_{p'/q'}),$$

then

$$p=a_ip'+b_iq',\qquad q=c_ip'+d_iq'$$

for some $1\leq i\leq k$. This is the claimed one-cusped consequence of the preceding semialgebraic conjecture and the Pila–Wilkie theorem; it remains unproved in the supplied text.