Nonsingularity of superelliptic Legendre inflectionary curves away from the singular fibers

Let $F$ be the base field, let $n\geq 2$, and let $a,b,c$ be positive integers. Let $\mathcal{C}^{\ell}_m$ be the inflectionary curve derived from the pencil $y^n=x^a(x-1)^b(x-\lambda)^c$, with distinguished points $p_1,p_2,p_3$. Generalized Legendre nonsingularity conjecture. Suppose that $\operatorname{char}(F)$ is zero or sufficiently positive. For all $\ell$ and $m$, the curve $\mathcal{C}^{\ell}_m$ is nonsingular away from $p_1,p_2,p_3$. This generalizes the cited conjecture for Legendre inflectionary curves; the assertion concerns the absence of additional singularities and remains unverified in the supplied text.