Maximal-dimension conjecture for constant-rank affine spaces of nilpotent matrices

Let $n\in\mathbb{N}-\{0\}$ and let $K$ be a field. Let $S$ be an affine subspace of $M(n\times n,K)$ such that every element of $S$ is nilpotent, and assume that each element of $S$ has rank equal to $r\geq 1$. Constant-rank nilpotent dimension conjecture. If $|K|$ is sufficiently large, then the maximal dimension $S$ can attain is

$$\sum_{i=1}^r(n-1-i)=\frac{r}{2}(2n-r-3).$$

This predicts the sharp dimension bound for affine spaces consisting of nilpotent matrices of fixed rank, extending the dimension question beyond the rank-one case and subject to a sufficiently large-field hypothesis.