Coefficient factorization conjecture for Weierstrass atomic inflection polynomials

Let $n=2\ell$, assume that the characteristic of $F$ is zero or sufficiently positive, and let $P^{\ell,*}_m$ be the atomic inflection polynomial of the Weierstrass pencil $y^n=x^3+\lambda x+2$, in coordinates centered at $(1,-3)$. Write $[(i,j)]P^{\ell,*}_m$ for the coefficient of the corresponding monomial, and let $(u)_r$ and $((u))^r$ denote the factorial-type factors used in the source. Weierstrass coefficient factorization conjecture. For $m=2k$, $k\geq3$, and $j=1,\ldots,k-2$, the stated coefficients equal $c_{j,k}(u)_k((2u-2k+1))^j$, with $c_{j,k}=\frac{3^{j+k}(2j+1)}{(k-j)!\prod_{i=1}^j i(2i+1)}$; for $m=2k+1$, they equal $d_{j,k}(u)_{k+1}((2u-2k+1))^j$, with $d_{j,k}=\frac{2\cdot3^{j+k+1}}{(k-j)!\prod_{i=1}^j i(2i+1)}$. The preceding discussion presents these coefficient formulas as an expected pattern, and no proof or resolution is supplied.