Singularity conjecture for D4 inflectionary curves

Let $n=2\ell$, assume that $\operatorname{char}(F)$ is zero or sufficiently positive, and let $\mathcal{C}_m=\mathcal{C}^{\ell}_m$ be the $m$-th inflectionary curve from the $D_4$ pencil $y^n=x^5+x^3+sx$, compactified in $\mathbb{P}(1,4,1)$. D4 singularity conjecture. For every $m\geq3$, $\mathcal{C}_m$ is geometrically irreducible and has precisely three singularities in the affine $xs$-plane, at $(0,0)$ and $(\pm\sqrt{-1/2},1/4)$; the latter two are exchanged by an involution and hence are isomorphic. The predicted points correspond to the singular points of the total space of the pencil; the supplied text gives no proof of the full assertion.