The GK-pair conjecture for good symmetric pairs

Let $(G,H)$ be a good symmetric pair, in the sense defined in the paper; thus $H=G^\theta$ for an involution of a reductive group $G$ and the additional goodness conditions of the paper hold.

GK-pair conjecture for good pairs. Every good symmetric pair is a GK-pair.

The paper presents this as a stronger statement implying the connected complex GK-pair conjecture. By the paper's theorem on regular symmetric pairs, it would follow from the conjecture that every symmetric pair is regular.