SL(2, C)-twisted homology product conjecture for taut sutured manifolds

Let $M=(M,R_-,R_+,\gamma)$ be a taut sutured manifold. A representation $\alpha\colon\pi_1(M)\to\operatorname{SL}_2(\mathbb C)$ makes $M$ a homology product when the inclusion-induced maps on homology with coefficients in the associated local coefficient system are isomorphisms. SL(2, C)-twisted homology product conjecture. There exists a representation

$$\alpha\colon\pi_1(M)\to\operatorname{SL}_2(\mathbb C)$$

for which $M$ is a homology product. This is presented as a weaker, less geometric parallel to the hyperbolic-knot conjecture. The source gives no resolution; it contrasts this claim with the known existence of some representation into a general general linear group.