The residue-class description of order-two kinematic varieties

Let $d$ and $n$ be positive integers, and let $\mathcal{K}^{(2)}_{d,n}$ denote the kinematic variety of order-two spinor brackets, represented by an $n\times n$ matrix. Write $\lfloor d/2\rfloor$ for the integer part of $d/2$.

The residue-class description. For $d \equiv 0,1,6,7 \pmod{8}$, the kinematic variety $\mathcal{K}^{(2)}_{d,n}$ consists of all symmetric $n\times n$ matrices with zero diagonal and rank at most $2^{\lfloor d/2\rfloor}$. For $d \equiv 2,3,4,5 \pmod{8}$, $\mathcal{K}^{(2)}_{d,n}$ is the variety of skew-symmetric $n\times n$ matrices of rank at most $2^{\lfloor d/2\rfloor}$.

This would give a uniform description of the order-two kinematic varieties in every spacetime dimension, extending the explicitly established low-dimensional cases. The supplied text does not state whether the claim has been proved or remains open.