Ramakrishnan's conjecture relating complex volume and the Bloch invariant

Let $\mathcal{M}$ and $\mathcal{N}$ be hyperbolic $3$-manifolds. Write $\operatorname{vol}_{\mathbb{C}}$ for complex volume and $\beta$ for the Bloch invariant. Ramakrishnan's conjecture. If

$$\operatorname{vol}_{\mathbb{C}}\mathcal{M}=\operatorname{vol}_{\mathbb{C}}\mathcal{N},$$

then

$$\beta(\mathcal{M})=\beta(\mathcal{N}).$$

The conjecture predicts that complex volume determines the Bloch invariant. The paper presents Dehn-filling results intended to shed light on it, but does not prove the conjecture.