Fourier-compatibility conjecture for Lie algebra smooth transfer

Let $\mathfrak s$ and $\mathfrak s'$ be the tangent spaces of the relevant symmetric spaces, equipped with the smooth-transfer relation defined by the transfer factor $\kappa$, and let $\widehat{f}$ denote Fourier transform. Fourier-compatibility conjecture. There exists a nonzero constant $c\in\mathbb C$ such that, if $f\in{\mathcal C}_c^\infty(\mathfrak s)$ is a smooth transfer of $f'\in{\mathcal C}_c^\infty(\mathfrak s')$, then $\widehat f$ is a smooth transfer of $c\widehat{f'}$. This conjecture is the Lie-algebra Fourier-transform step in the reduction of group smooth transfer and is presented as an open ingredient of the paper's method.