Pseudo complex volume uniqueness conjecture for hyperbolic Dehn fillings

Let $\mathcal{M}$ be an $n$-cusped hyperbolic $3$-manifold with non-symmetric cusp shapes. Let $N_{\mathcal{M}}^{\mathbb{C}}(v)$ denote the number of Dehn fillings of $\mathcal{M}$ whose pseudo complex volume is $v$. Pseudo complex volume uniqueness conjecture. For every $v\in\mathbb{C}$ sufficiently close to $\operatorname{vol}_{\mathbb{C}}\mathcal{M}$ modulo $i\pi^2\mathbb{Z}$, one has

$$N_{\mathcal{M}}^{\mathbb{C}}(v)=1.$$

The preceding theorem establishes uniqueness for sufficiently large filling coefficients; the conjecture asserts the corresponding local uniqueness near the complex volume of the unfilled manifold without explicitly imposing that coefficient bound.