Feigin–Wang–Yoshinaga conjecture for the extended Catalan arrangement of type $B_2$

Let $m,i\in\mathbb{Z}_{\geq 0}$, and let $\widetilde{f_i^m}(x,y)$ be the deformed polynomial defined from the coefficients of

$$f_i^m(x,y)=\int_0^x t^{2i}(t^2-x^2)^m(t^2-y^2)^m\,dt.$$

For a polynomial $g$, write $g\in((x+y+m)_{2m+1})$ when it is divisible by the falling factorial $(x+y+m)_{2m+1}$. Feigin–Wang–Yoshinaga conjecture. For $i,m\in\mathbb{Z}_{\geq 0}$,

$$\widetilde{f_i^m}(x+y)+\widetilde{f_i^m}(x+y)\in((x+y+m)_{2m+1}).$$

This divisibility assertion is the remaining condition needed to show that the deformed logarithmic vector fields form a basis for the module associated with the extended Catalan arrangement $\operatorname{Cat}(B_2,m)$. The source presents it as a conjecture of Feigin, Wang and Yoshinaga; its resolution is not specified here.