For physics application, I want to calculate the integral of the form
$$I(z)=\int_{0}^{\infty}\int_{0}^{\infty}H^{1,1}_{2,2}(x)H^{1,1}_{2,2}(y)H^{1,1}_{2,2}(z+\lambda x +y)H^{1,1}_{2,2}(z-x-\lambda y) \text{ d}x\text{ d}y$$
where $H^{1,1}_{2,2}(\xi)$ denotes the Fox H-function:
$H^{1,1}_{2,2}(\xi)=H^{1,1}_{2,2} \left[ \xi \Bigg| \begin{matrix} (1-\beta,\beta), & (1/2+\epsilon,1/2-\epsilon) \\ (0,1), & (1/2+\epsilon,1/2-\epsilon) \end{matrix}\right].$
Can this integral be calculated somehow analytically? Can the result be written in terms of some Fox H-function?