van Dijk's GK-pair conjecture for connected complex symmetric pairs

Let $F=\mathbb{C}$, and let $(G,H)$ be a connected symmetric pair over $F$. A GK pair is a symmetric pair satisfying the Gelfand–Kazhdan distributional invariance criterion used in the paper.

van Dijk's GK conjecture. If $F=\mathbb{C}$, any connected symmetric pair is a GK pair.

The paper states that this conjecture implies van Dijk's Gelfand-pair conjecture by the distribution criterion. It is then implied by the more general conjecture that every good symmetric pair is a GK pair; no resolution is supplied.