Finite coordinatewise parametrization conjecture for equal-volume multi-cusped fillings

Let $\mathcal{M}$ be an $n$-cusped hyperbolic $3$-manifold, and let $\mathcal{M}_{(p_1/q_1,\dots,p_n/q_n)}$ denote the corresponding hyperbolic Dehn filling. Finite coordinatewise parametrization conjecture. There exists a collection

$$\{(a_{i,j},b_{i,j},c_{i,j},d_{i,j})\in\mathbb{Z}^4\}_{(1\leq i\leq n,\ 1\leq j\leq n_i)}$$

such that if

$$\operatorname{vol}\bigl(\mathcal{M}_{(p_1/q_1,\dots,p_n/q_n)}\bigr)=\operatorname{vol}\bigl(\mathcal{M}_{(p'_1/q'_1,\dots,p'_n/q'_n)}\bigr),$$

then

$$p_i=a_{\sigma(i),j}p'_{\sigma(i)}+b_{\sigma(i),j}q'_{\sigma(i)},\qquad q_i=c_{\sigma(i),j}p'_{\sigma(i)}+d_{\sigma(i),j}q'_{\sigma(i)}$$

for some $\sigma\in S_n$ and $1\leq j\leq n_{\sigma(i)}$. This is proposed as the multi-cusped analogue of the one-cusped parametrization and is not proved in the supplied text.