The conjecture for the maximal dimension of constant-rank matrix spaces

Let $k$ be an algebraically closed field of characteristic zero, and let $l(r;a)$ be the maximal dimension of a subspace of $a\times a$ matrices such that every nonzero matrix has rank $r$. Assume that $a$ and $r$ are integers satisfying

$$\frac{2a+2}{3}>r>\frac{a}{2}+1.$$

Constant-rank matrix dimension conjecture. Then

$$l(r;a)=a-r+1.$$

This conjecture is obtained by translating the uniform-bundle conjecture into the language of constant-rank matrix spaces. The paper verifies it for all $a\leq 10$, while the stated range is not settled in general.