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

Let $F=\mathbb{C}$, and let $(G,H)$ be a connected symmetric pair over $F$, with $G/H$ connected. A Gelfand pair is a pair for which the relevant representation-theoretic multiplicity-one property holds.

van Dijk's conjecture. Any connected symmetric pair $(G,H)$ over $\mathbb{C}$ such that $G/H$ is connected is a Gelfand pair.

This is the paper's stated van Dijk conjecture. The surrounding discussion explains that it would follow from the conjecture that every symmetric pair is regular, while no resolution is given here.