The GK-pair conjecture for connected complex symmetric pairs

Let $F=\mathbb{C}$ and let $(G,H)$ be a connected symmetric pair, with $H=G^\theta$ for an involution $\theta$ of a reductive group $G$. A pair is a GK-pair when it satisfies the Gelfand–Kazhdan property defined in the paper.

GK-pair conjecture. Any connected symmetric pair over $\mathbb{C}$ is a GK-pair.

The paper states that this stronger conjecture would imply van Dijk's Gelfand-pair conjecture. It in turn would follow from the conjecture that every good symmetric pair is a GK-pair, and hence from the regularity conjecture for all symmetric pairs.