Semialgebraic structure conjecture for equal-volume Dehn fillings

Let $\mathcal{M}$ be a $1$-cusped hyperbolic $3$-manifold with volume function $\Theta_{\mathcal{M}}(x,y)$, and let $\mathcal{Z}$ be the analytic variety defined by

$$\Theta_{\mathcal{M}}(x,y)=\Theta_{\mathcal{M}}(z,w).$$

Let $Q_{\mathcal{M}}(x,y)$ denote the quadratic form associated with the cusp of $\mathcal{M}$. Semialgebraic structure conjecture. Every semialgebraic subset of $\mathcal{Z}$ is contained in

$$z=ax+by,\qquad w=cx+dy$$

for some $a,b,c,d\in\mathbb{Z}$ satisfying

$$Q_{\mathcal{M}}(x,y)=Q_{\mathcal{M}}(ax+by,cx+dy).$$

This conjecture describes the semialgebraic loci responsible for coincidences among volumes of Dehn fillings. Together with the Pila–Wilkie theorem and the finiteness of the relevant integer quadruples, it is intended to control equal-volume fillings, but no proof is given here.