Szőnyi's minimum-size conjecture for non-trivial blocking sets

Let $p$ be a prime. Suppose that $\mathbb{F}_{p^e}$ is the maximum field of linearity of a non-trivial blocking set $B$ in $\operatorname{PG}(2,p^t)$, where $t=en$. A blocking set is a set of points meeting every line, and it is non-trivial if it contains no line.

Szőnyi's conjecture. The blocking set $B$ has at least

$$(p^e)^n+(p^e)^{n-1}+1$$

points.

This conjecture concerns the minimum size of non-trivial blocking sets and the role of their largest field of linearity. The supplied text gives no resolution status, so the conjecture is recorded as open.