Nil-projection conjecture for double flag varieties

Let $\mathfrak{X}$ be a double flag variety of finite type, with projection

$$\phi_{\mathfrak{s}}:\mathcal{Y}\to\mathfrak{s}.$$

Let $\mathcal{N}_{\mathfrak{s}}$ denote the nilpotent variety in $\mathfrak{s}$. Nil-projection conjecture. The image of $\phi_{\mathfrak{s}}$ is contained in $\mathcal{N}_{\mathfrak{s}}$.

No counterexample is known, and the claim holds in the cases established by the cited propositions; its validity for all double flag varieties of finite type remains open.