Melham's conjecture on odd power sums of Fibonacci numbers

Let $F_n$ denote the $n$-th Fibonacci number and $L_n$ the $n$-th Lucas number, with $F_0=0$, $F_1=1$, $L_0=2$, and $L_1=1$. For $n\geq 2$, they satisfy

$$F_n=F_{n-1}+F_{n-2},\qquad L_n=L_{n-1}+L_{n-2}.$$

Melham's conjecture. For any positive integers $n,m$, there is a polynomial $P_{2m-1}(x)$ of degree $2m-1$ with integer coefficients such that

$$L_1L_3L_5\cdots L_{2m+1}\sum_{k=1}^n F_{2k}^{2m+1}=(F_{2n+1}-1)^2P_{2m-1}(F_{2n+1}).$$

The conjecture concerns an integral polynomial representation for odd power sums of even-indexed Fibonacci numbers. The paper states that it proves this conjecture by showing the relevant polynomial and its derivative vanish at $1$, and that multiplying by the indicated product of odd-indexed Lucas numbers yields an integer polynomial.