The stable-range theta lift low-rank conjecture

Let $(G',G)$ be a dual pair in the stable range, with $G'$ the smaller member. Let $\tau$ be an irreducible representation of $G$.

Stable-range theta lift low-rank conjecture. The theta lift $\theta(\tau)$ is a low rank representation of $G$, meaning that its $U$-support contains no open $L$-orbit.

The conjecture predicts that stable-range theta lifts from the smaller member of a dual pair give low-rank representations. The supplied status evidence indicates that the paper treats the surrounding question as open.