Bounded-support conjecture for the restricted Weyl group action

Let $\mathfrak{g}$ be any simple real Lie algebra of rank $r$, and let $\lambda\in Q\cap\mathfrak{h}^+$ be a dominant weight, written in the basis of fundamental weights as

$$\lambda=\sum_{i=1}^r c_i\varpi_i.$$

Suppose that the space $V_\lambda^\mathfrak{l}$ of $\mathfrak{l}$-invariant vectors is nonzero. Bounded-support conjecture. The restriction of the longest Weyl group element satisfies

$${\left.w_0\right|}_{V_\lambda^\mathfrak{l}}=\pm\operatorname{Id}$$

only when at most $3$ of the coordinates $c_i$ are nonzero. This conjecture generalizes the corresponding statement in the split case, where the bound is $1$. The evidence described in the source is experimental, based on computations for sufficiently low-dimensional representations and sufficiently low-rank simple Lie algebras; the general assertion remains open.