The irrationality conjecture for peripheral translation parameters of hyperbolic knots

Let $K$ be a hyperbolic knot in $S^3$, let $R_0$ be the relevant irreducible component of its $SL_2(\mathbb{C})$-representation variety, and let $\mu$ and $\lambda$ denote the meridian and longitude. Suppose $\rho\in R_0$ is an irreducible representation such that, after conjugation,

$$\rho(\mu)=\pm\begin{bmatrix}1&1\\0&1\end{bmatrix},\qquad \rho(\lambda)=\pm\begin{bmatrix}1&t(\rho)\\0&1\end{bmatrix},$$

where $t(\rho)\in\mathbb{C}$. Irrationality conjecture. If $\rho(\mu)$ and $\rho(\lambda)$ are parabolic, then $t(\rho)\notin\mathbb{Q}$. This asserts that the peripheral translation parameter of such a representation cannot be rational; the source provides no resolution, so the conjecture remains open.