Dunfield's orderable Dehn filling conjecture for two-bridge knots

Let $K$ be a two-bridge knot, and suppose that its Alexander polynomial has a root $p_0$. Let $S_1$ and $S_2$ be slopes of incompressible surfaces associated to ideal points of the $\operatorname{PSL}_2\mathbb{C}$ character variety of the complement of $K$. For some odd integer $k$ satisfying $0<k\leq 2g(K)-1$, define

$$I=\begin{cases} (-S_1,-S_2) & \text{if }p_0>0,\\ (-\infty,k) & \text{if }p_0\text{ is a unit complex number}. \end{cases}$$

Dunfield's conjecture. For every rational slope $r\in I$, Dehn filling $K$ along slope $r$ is orderable.

This conjecture connects orderability of Dehn fillings of two-bridge knots with slopes arising from ideal points of the character variety. The source presents it as a conjecture motivated by the observed relationship between holonomy extension loci and incompressible-surface boundary slopes; its resolution status is not specified here.