Generic rank-one certifying representations for taut genus-two handlebodies

Let $M$ be a taut sutured genus-two handlebody. A representation $\alpha:pi_1(M)\to\operatorname{GL}_1(\mathbb C)$ is certifying when the inclusion-induced maps

$$H_*(R_\pm;E_\alpha)\to H_*(M;E_\alpha)$$

are isomorphisms. The generic rank-one certification conjecture. The representations

$$\alpha:\pi_1(M)\to\operatorname{GL}_1(\mathbb C)$$

which certify $M$ as an $\alpha$-homology product form a non-empty, Zariski open subset of the one-dimensional representation variety

$$\operatorname{Hom}(\pi_1(M),\operatorname{GL}_1(\mathbb C)).$$

The conjecture extends the result that the specific example in the preceding theorem is certified on a non-empty Zariski open set, and would establish generic rank-one certification for all taut sutured genus-two handlebodies.