The $P(\mathfrak g,e)$ conjecture for split semisimple Lie algebras

Let $\mathfrak g$ be a split semisimple real Lie algebra and let $e\in\mathfrak g$ be nilpotent. Denote by $P(\mathfrak g,e)$ the statement that, whenever $E\subset\mathfrak g_{\mathbb C}$ is a subalgebra satisfying

$$Z_{\mathfrak g_{\mathbb C}}(e)\subset E,\qquad E+\overline E=\mathfrak g_{\mathbb C},$$

then $E=\mathfrak g_{\mathbb C}$. $P(\mathfrak g,e)$ conjecture. $P(\mathfrak g,e)$ is true for every split semisimple Lie algebra $\mathfrak g$ and every nilpotent element $e\in\mathfrak g$. This condition is used to extend the classification of generalized complex structures from nilpotent orbits in $\mathfrak{sl}_n(\mathbb R)$ to arbitrary split semisimple Lie algebras; the source leaves its general validity open.