On additive MDS codes with linear projections

We support some evidence that a long additive MDS code over a finite field must be equivalent to a linear code. More precisely, let $C$ be an ${\mathbb{F}}_q$-linear $(n,q^{hk},n-k+1{)}_{q^h}$ MDS code over ${\mathbb{F}}_{q^h}$. If $k=3$, $h\in \{2,3\}$, $n>\max \{q^{h-1},hq-1\}+3$, and $C$ has three coordinates from which its projections are equivalent to linear codes, we prove that $C$ itself is equivalent to a linear code. If $k>3$, $n>q+k$, and there are two disjoint subsets of coordinates whose combined size is at most $k-2$ from which the projections of $C$ are equivalent to linear codes, we prove that $C$ is equivalent to a code which is linear over a larger field than ${\mathbb{F}}_q$.