The maximal affine-set linearity conjecture

Let $\mathcal{U}\subseteq\operatorname{AG}(2,q)$ be a maximal affine set, let $\mathcal{D}$ be the set of directions determined by $\mathcal{U}$, and let $s$ and $t$ be the associated parameters.

Maximal affine-set linearity conjecture. If $t=s>2$, then $\mathcal{U}$ is an affine $\operatorname{GF}(s)$-linear set.

The conjecture proposes linearity for maximal affine sets in the case $s=t>2$, following examples of maximal affine sets that are not linear when $s=1$. The supplied text gives no resolution, so the conjecture remains open.