Type C fourth-order ODE conjecture for superelliptic polynomial families

Let $r\geq 3$ and let $j_0\in\{-r+1,\ldots,-2\}$. For each such initial condition, let $\{P_{j_0,n}(c)\}$ be the corresponding polynomial family. Type C ODE conjecture. For each $j_0\in\{-r+1,\ldots,-2\}$, the polynomial family $\{P_{j_0,n}(c)\}$ satisfies a fourth-order linear ODE in $c$, derivable by the same Gegenbauer elimination strategy as types 1 and 2. These initial conditions produce genuinely new polynomial families whose generating functions involve mixed superelliptic integrals; the conjectured ODE remains to be derived.