Irreducibility conjecture for the bivariate Hahn polynomial numerator

Let $n\in\mathbb{N}$ and let $\ell_1,\ell_2\in\mathbb{N}$ satisfy

$$n\leq \ell_1\wedge\ell_2.$$

Define the polynomial

$$R_n(x;\ell_1,\ell_2,y):=\sum_{k=0}^{n}\frac{(-n)_k(n-\ell_1-\ell_2-1)_k(-x)_k(-y+k)_{n-k}}{(-\ell_1)_k\,k!}.$$

Irreducibility conjecture. The polynomial $R_n(x;\ell_1,\ell_2,y)$ is irreducible unless $\ell_1=\ell_2$ and $n$ is odd. In that exceptional case, $R_n(x;\ell_1,\ell_2,y)$ is the product of $y-2x$ and an irreducible polynomial of degree $n-1$. This conjecture asserts that the displayed linear factor is the only nontrivial factor in the exceptional case; the factor is explained by the symmetry identity relating $R_n(x;\ell_1,\ell_2,y)$ to $R_n(y-x;\ell_2,\ell_1,y)$.