Orthogonal Plücker-type conjecture for simple forms

Let $\mathbb{V}$ be a finite-dimensional real vector space with a euclidean or lorentzian inner product, let $p\geq 2$, and let $F\in\Lambda^p\mathbb{V}^*$. For $\Xi\in\Lambda^{p-2}\mathbb{V}$, write $\iota_\Xi F$ for the resulting $2$-form and let $[\iota_\Xi F,F]$ denote its induced action on $F$. Orthogonal Plücker-type conjecture. (i) If $\dim\mathbb{V}=d=2p$ or $d=2p+1$, then, for all $\Xi\in\Lambda^{p-2}\mathbb{V}$,

$$[\iota_\Xi F,F]=0$$

if and only if $F$ is a sum of two orthogonal simple forms:

$$F=F_1+F_2,$$

where $F_1$ and $F_2$ are simple and $F_1\perp F_2$. (ii) If $p\leq d<2p$, then the same equation holds if and only if $F$ is simple. These conditions are motivated by the Plücker relations for forms and by their role in supersymmetric supergravity solutions. The first part had been verified in several low-dimensional cases listed in the source, while the second part had also been verified in selected cases; the general assertions remain conjectural in the stated signatures and dimensions.