Buttcane's conjecture on the interchange of integral

Let $G(\mathbb{R})$ be the real group under consideration, let $U(\mathbb{R})$ be its maximal unipotent subgroup, and let $U_\sigma(\mathbb{R})$ be the subgroup associated with a relevant Weyl-group element $\sigma$. Write $\mathfrak{a}^*$ for the relevant real vector space, $\mathfrak{a}^*_{\mathbb{C}}$ for its complexification, and let $W(-i\nu,\cdot,\psi)$ be the Whittaker function for a generic character $\psi$. For $h$ smooth and compactly supported with $h(0)=1$, define the regularized kernel by

$$\widetilde K_\sigma(-i\nu,y,t)=\lim_{R\to0}\int_{U_\sigma(\mathbb{R})\backslash U(\mathbb{R})}h\left(\frac{\lVert u_1\rVert}{R}\right)W(-i\nu,y\sigma u_1t,\psi)\overline{\psi(u_1)}\,du_1.$$

Buttcane's conjecture. Let $g$ be holomorphic with rapid decay on an open tube domain in $\mathfrak{a}^*_{\mathbb{C}}$ containing $\mathfrak{a}^*$, let $t\in G(\mathbb{R})$, let $\psi$ be generic, and let $\sigma$ be relevant. Then for almost all $y\in\operatorname{Sp}_4(\mathbb{R})$ satisfying $yuy^{-1}u^{-1}\in U_0(\mathbb{R})$ for all $u\in U_\sigma(\mathbb{R})$,

$$\int_{U_\sigma(\mathbb{R})\backslash U(\mathbb{R})}\int_{\mathfrak{a}^*}g(-i\nu)W(-i\nu,y\sigma u_1t,\psi)\,d\nu\,\overline{\psi(u_1)}\,du_1=\int_{\mathfrak{a}^*}g(-i\nu)\widetilde K_\sigma(-i\nu,y,t)\,d\nu.$$

Moreover, $\widetilde K_\sigma$ is entire in $\nu$ and smooth and polynomially bounded in $t$ and $y$ when $\operatorname{Re}(-i\nu)$ lies in a fixed compact set. This interchange would justify removing the $\mathfrak{a}^*$-integration from the preceding Archimedean orbital-integral formula, while the asserted regularity and bounds describe the analytic control needed for that application.