Detection of all $S_4(q,r)$ surfaces by the two-dimensional character-variety component

Let $\mathbb{F}$ be an algebraically closed field of characteristic $p$. Consider $N_\Phi$ a Seifert fibered space with base orbifold $S^2(2,2,2,3)$ and gluing matrix $\Phi$ $=$

$$\begin{pmatrix} k & \pm1 \\ \mp1-6k & \mp6 \end{pmatrix}\in\operatorname{SL}_2(\mathbb{Z}).$$

There is a curve in $X(N_\Phi,\mathbb{F})$ that detects $S_4(q,r)$. Specifically, $S_4(q,r)$ is detected by $C(q-r,r)$ given in.

The conjecture extends the preceding result for $S_4(q,1)$ to all surfaces $S_4(q,r)$ using the two-dimensional component of the character variety. The source reports additional calculations in this direction, but does not establish the claim.