Divisibility conjecture for the deformed polynomials of type B2

Let $m,i\geq 0$, and let $\widetilde{f^m_i}(x,y)$ denote the deformed polynomial defined above for the type $B_2$ construction. The relevant derivation condition requires divisibility by the product of the linear forms $x+y-k$ for $-m\leq k\leq m$.

Divisibility conjecture. For any $m,i\geq 0$, the polynomial $\widetilde{f^m_i}(x,y)+\widetilde{f^m_i}(y,x)$ is divisible by

$$\prod_{k=-m}^m(x+y-k).$$

This assertion supplies the remaining divisibility condition needed for the displayed functions to define derivations in the type $B_2$ multiarrangement setting. The supplied text gives examples satisfying the conditions but does not state a proof or resolution of this general claim.