The pseudo complex volume–complex volume equivalence conjecture for Dehn fillings

Let $\mathcal{M}$ be an $n$-cusped hyperbolic $3$-manifold. Let $\mathcal{M}_{p_1/q_1, \dots, p_n/q_n}$ and $\mathcal{M}_{p'_1/q'_2,\dots, p'_n/q'_n}$ be two Dehn fillings of $\mathcal{M}$. The pseudo complex volume–complex volume equivalence conjecture. Their pseudo complex volumes agree if and only if their complex volumes agree:

$$\operatorname{pvol}_{\mathbb{C}}\mathcal{M}_{p_1/q_1, \dots, p_n/q_n}=\operatorname{pvol}_{\mathbb{C}}\mathcal{M}_{p'_1/q'_2,\dots, p'_n/q'_n}$$

if and only if

$$\operatorname{vol}_{\mathbb{C}}\mathcal{M}_{p_1/q_1, \dots, p_n/q_n}=\operatorname{vol}_{\mathbb{C}}\mathcal{M}_{p'_1/q'_1,\dots, p'_n/q'_n}.$$

The conjecture proposes that classifying sufficiently general Dehn fillings by pseudo complex volume is equivalent to classifying them by complex volume; the supplied text gives supporting results for two-cusped fillings but no resolution of the general statement.