The conjectural double Fourier–Mellin transform formula over the complex numbers

Let $p$ be an integer, let $\beta,\gamma,\nu\in\mathbb{C}$, and let $u,\varw\in\mathbb{C}$ satisfy

$$\operatorname{Re}(1/(u\varw))<1.$$

For the Bessel function $\boldsymbol{B}_{\nu,p}$ and the associated function $\boldsymbol{H}_{\nu,p}^{\beta-\gamma}$, the double Fourier–Mellin transform conjecture. The integral formula

$$\begin{aligned} &\int\!\!\int\!\!\int\!\!\int \boldsymbol{B}_{\nu,p}(\sqrt{\varv z})e\bigl(-2\operatorname{Re}(uz+\varw\varv)\bigr)\frac{i\,dz\wedge d\widebar{z}}{|z|^{2-2\beta}}\frac{i\,d\varv\wedge d\widebar{\varv}}{|\varv|^{2-2\gamma}} \\ &\qquad=\frac{2}{(2\pi)^{2\beta+2\gamma}|u|^{2\beta}|\varw|^{2\gamma}}\,{}^{\beta+\gamma}\!\boldsymbol{H}_{\nu,p}^{\beta-\gamma}\!\left(\frac{1}{u\varw}\right) \end{aligned}$$

is valid whenever

$$|\operatorname{Re}(\nu)|<\operatorname{Re}(2\beta)<\frac{3}{2},\qquad |\operatorname{Re}(\nu)|<\operatorname{Re}(2\gamma)<\min\bigl\{1+\operatorname{Re}(2\beta),3-\operatorname{Re}(2\beta)\bigr\}.$$

This conjecture extends the preceding complex double Fourier–Mellin identity from even $p$ to arbitrary $p$ and general Mellin exponents. Its validity is motivated by the corresponding real-variable formulas, but the supplied text gives no proof or resolution.