The nilradical containment conjecture for constant-length Killing fields

Let $(G/H,g)$ be a geodesic orbit Riemannian space, let $\mathfrak{g}=\operatorname{Lie}(G)$, and let $X\in\mathfrak{g}$ be a Killing field of constant length. Define

$$A_1=\operatorname{Ker}(L^2),\qquad A_2=\operatorname{Im}(L^2),$$

where $L=\operatorname{ad}(X)$. Let $\mathfrak{n}(\mathfrak{g})$ denote the nilradical of $\mathfrak{g}$. The nilradical containment conjecture. If $X\in\mathfrak{g}$ has constant length on $(G/H,g)$, then

$$\mathfrak{n}(\mathfrak{g})\subset A_1.$$

The nilradical of the Lie algebra of a geodesic orbit space is known to be commutative or two-step nilpotent, and the stated containment is presented as a consequence suggested by the preceding structural results; no resolution is supplied here.