The factorization conjecture for the two-variable Bernoulli determinant

Let $r\geq 1$, and let $\bar F_{r,2}(x_1,x_2)$ be the determinant polynomial defined in the paper for $D=2$. Factorization conjecture. One has

$$\bar F_{r,2}(x_1,x_2)=(-1)^{\binom{r}{2}} \frac{[1!2!\cdots(r-1)!]^3}{r!(r+1)!\cdots(2r-1)!}(x_2-x_1)^r \prod_{j=0}^{r-1}((x_2-x_1)^2-j^2)^{r-j}.$$

The formula was supported by computer experiments and checked in the source for $r\leq 4$, but no proof for all $r$ is given.