The AGS equivalence conjecture for complex semisimple Lie algebra representations

Let $\mathfrak h$ be a complex semisimple Lie algebra and $V$ an orthogonal representation of $\mathfrak h$ over $\mathbb C$. AGS equivalence conjecture. One has

$$\rho_{\mathfrak h}\leq\rho_V\quad\Longleftrightarrow\quad V\text{ has AGS in }\mathfrak h.$$

This is the complex form of the preceding real conjecture and is stated to be equivalent to it using the proposition and the lemma on complexification. It is known in the settings listed for the real version, while the general complex case remains open.