The AGS conjecture for real semisimple Lie algebra representations

Let $\mathfrak h$ be a real semisimple Lie algebra and $V$ an orthogonal representation of $\mathfrak h$. AGS conjecture. If $V$ has AGS in $\mathfrak h$, then

$$\rho_{\mathfrak h}\leq\rho_V.$$

The conjecture gives a converse-type criterion to the implication in the preceding proposition. It is stated for arbitrary real semisimple $\mathfrak h$ and is known when $\mathfrak h$ is simple, or when $V=\mathfrak g/\mathfrak h$ for a semisimple Lie algebra $\mathfrak g$ containing $\mathfrak h$ as a subalgebra; the general case remains open.