Riley's real-root conjecture for parabolic representations of two-bridge knots

Let $K$ be a two-bridge knot, and let $\Phi_K(2,y)$ be its Riley polynomial specialized at meridian trace $2$. The knot signature is denoted by $\sigma(K)$. Riley's conjecture. The equation

$$\Phi_K(2,y)=0$$

has at least $\frac{1}{2}|\sigma(K)|$ real solutions. This conjecture relates the number of real parabolic $SL_2(\mathbb C)$ representations of a two-bridge knot group to the knot signature. The source gives no resolution beyond proving the conjecture for double twist knots.