Generation-degree conjecture for the square variety of skew-symmetric matrices

Let $X$ be a real skew-symmetric $n\times n$ matrix, and let $\mathcal{V}_{X^2}$ denote the variety of symmetric matrices arising as squares $X^2$. Generation-degree conjecture. The prime ideal of the variety $\mathcal{V}_{X^2}$ is generated in degree $\lceil n/2\rceil$. The low-dimensional examples suggest this degree pattern, but no general proof or resolution is given here.