A linear-in-$q$ range of dimensions for the MDS conjecture over $\mathbb F_q$ in odd characteristic

Let $q$ be a power of an odd prime $p$. We prove the MDS conjecture over $\mathbb F_q$ in dimension $k$ whenever \[ 2\leqslant k\leqslant B(p,q) \quad\text{or}\quad q+2-B(p,q)\leqslant k\leqslant q, \] where $B(p,q)=\left\lfloor\frac{(p-2)q+6p-10}{2p-3}\right\rfloor$. For fixed $p$, this gives