Polynomial representation conjecture for the odd-power binomial sums

Let $a,b,c$ be non-negative integers and, for $r\in\mathbb{Z}^+$, define

$$S_r(a,b,c):=\sum_{k=0}^a k^{2r+1}\binom{a+b}{a+k}\binom{b+c}{b+k}\binom{c+a}{c+k}.$$

Let $\mathcal{L}$ act on functions of $a,b,c$ by

$$\mathcal{L}\cdot f=[(a+b)(a+c)E-a^2I]f,$$

where $E\cdot f(a,b,c)=f(a-1,b,c)$ and $I\cdot f(a,b,c)=f(a,b,c)$. Polynomial representation conjecture. For each $r\in\mathbb{Z}^+$, there exist symmetric polynomials $f_r,g_r\in\mathbb{Z}[a,b,c]$ such that

$$S_r(a,b,c)=\frac{b^2c^2f_r(a,b,c)\binom{b+c}{b}}{2}\sum_{j=0}^{a-1}\frac{\binom{b+j}{b}\binom{c+j}{c}\,g_r(j+1,b,c)}{f_r(j,b,c)f_r(j+1,b,c)}.$$

The polynomials $f_r$ satisfy

$$f_r(a,b,c)=\mathcal{L}\cdot f_{r-1}(a,b,c),\qquad f_0(a,b,c)=1.$$

The claim proposes a structured symmetric-polynomial form for these binomial sums and recursively determines the denominator polynomials. The surrounding text presents it as a problem, and a postscript reports that Matthew Hongye Xie had found a proof; the database status is therefore left open because the supplied parser status is unknown.