The faithful representation version of Thurston's Euler class one conjecture

Let $M$ be a closed hyperbolic 3-manifold with infinite first homology, and let $a\in H^2(M;\mathbb{R})$ be an integral class satisfying the parity condition and having dual norm exactly one. A representation is faithful if it is injective. The representation conjecture. There is a faithful representation

$$\rho:\pi_1(M)\longrightarrow \operatorname{Homeo}^+(S^1)$$

whose Euler class is $a$. This is presented as a weaker version of Thurston's original conjecture, because representations into $\operatorname{Homeo}^+(S^1)$ need not arise from taut foliations; the source does not provide evidence of resolution.