Local Newton polygon conjecture for D4 inflectionary curves

Let $n=2\ell$, assume that $\operatorname{char}(F)$ is zero or sufficiently positive, and let $\mathcal{C}_m$ be the $m$-th inflectionary curve from $y^n=x^5+x^3+sx$. For $p=(\pm\sqrt{-1/2},1/4)$, let $\operatorname{New}_p(\mathcal{C}_m)$ be the Newton polygon in affine coordinates centered at $p$. D4 Newton polygon conjecture. The polygons are $\operatorname{Conv}((0,3),(0,2),(2,1),(5,0),(12,0))$ for $m=3$, $\operatorname{Conv}((0,4),(0,2),(2,1),(8,0),(16,0))$ for $m=4$, and $\operatorname{Conv}((0,5),(0,3),(1,2),(3,1),(9,0),(20,0))$ for $m=5$; for $m\geq6$ they have the displayed general form involving $\lceil m/2\rceil$ and the parity indicator. These predictions refine the expected local singularity types, but remain unproved in the supplied text.