Fourier positivity conjecture for spherical positive definite functions

Let $G$ be a semisimple linear group over a local field $F$, let $K<G$ be a maximal compact subgroup, and let $H<G$ be a closed abelian subgroup. Define

$$H_K:=H/(H\cap K).$$

For a continuous bi-$K$-invariant positive definite function $\psi$ on $G$ with $\psi|_H\in L^1(H_K)$, say that the pair $(H,\psi)$ has Fourier positivity when the Fourier transform of the corresponding function on $H_K$ is strictly positive on $\widehat H_K$. Fourier positivity conjecture. If $H<G$ is a closed non-compact abelian subgroup and $\psi$ is a continuous bi-$K$-invariant positive definite function on $G$ such that $\psi|_H\in L^1(H_K)$, then the pair $(H,\psi)$ has Fourier positivity.

This conjecture proposes strict positivity of Fourier transforms for restrictions of spherical positive definite functions to non-compact abelian subgroups. Its resolution is not established by the supplied source context.