Baer-subplane covering conjecture

Let $\Pi_q$ be an arbitrary projective plane of square order $q$, sufficiently large, and let $\Pi'$ be a Baer subplane of $\Pi_q$. Let $\mathcal{L}$ be a set of lines of $\Pi_q$ satisfying

$$|\mathcal{L}|\leq q+\sqrt{q}+1.$$

Baer-subplane covering conjecture. If $\mathcal{L}$ covers every point of $\Pi_q$ outside $\Pi'$, then $\mathcal{L}$ is a covering set; in particular, it also covers every point of $\Pi'$. This asks whether covering the complement of a Baer subplane with at most $q+\sqrt q+1$ lines necessarily forces coverage of the entire plane.