The $B_2$ eigenfunction conjecture for the series $f^{B_2}$

Let $s_1,s_2$ be generic parameters, and let $\lambda_1,\lambda_2$ satisfy $s_1=tT^{1/2}q^{\lambda_1}$ and $s_2=T^{1/2}q^{\lambda_2}$. Define the series $f^{B_2}(x_1,x_2;s_1,s_2,q,t,T)$ by the displayed expansion in the source, with coefficients $f^{B_2}_n$ and $c^{B_2}$ as specified there. Let $E_{\omega_1}(q,t,T)$ denote the operator used in the source. The $B_2$ eigenfunction conjecture. The series $f^{B_2}(x_1,x_2;s_1,s_2,q,t,T)$ satisfies

$$E_{\omega_1}(q,t,T)f^{B_2}(x_1,x_2;s_1,s_2,q,t,T)=tT^{1/2}(s_1+s_2+s_1^{-1}+s_2^{-1})f^{B_2}(x_1,x_2;s_1,s_2,q,t,T).$$

This asserts that the explicitly constructed $B_2$ series is an eigenfunction of the relevant difference operator, with eigenvalue determined by the parameters. The supplied text gives no resolution status or further context, so the claim is recorded as open pending verification.