Degree formula for realization spaces of matrix multiplication varieties

Let $\mathcal{M}_{r,m\times n}$ be the complex matrix multiplication variety and let $\mathcal{R}(\mathcal{M}_{r,m\times n})$ denote its realization space, viewed through its Zariski closure in $\mathbb{C}^{2m\times 2n}$. Then

Realization-space degree conjecture.

$$\deg \mathcal{R}(\mathcal{M}_{r, m \times n}) = (\deg \mathcal{M}_{r, m \times n})^2.$$

The conjecture would provide the degree formula needed for the real components of the equivariant autoencoder varieties, analogous to the degree formula established in the complex case. Its status is not resolved in the supplied text.