Primitive elements of finite fields $\mathbf{F}_{q^r}$ avoiding affine hyperplanes for $q=4$ and $q=5$

For a finite field $\mathbf{F}_{q^r}$ with fixed $q$ and $r$ sufficiently large, we prove the existence of a primitive element outside of a set of $r$ many affine hyperplanes for $q=4$ and $q=5$. This complements earlier results by Fernandes and Reis for $q\ge 7$. For $q=3$ the analogous result can be derived from a very recent bound on character sums of Iyer and Shparlinski. For $q=2$ the set consists only of a single element, and such a result is thus not possible.