  We show that a unital $U$ in $\mathrm{PG}(2,q^2)$ containing a point $P$,
such that at least $q^2-\epsilon$ of the secant lines through $P$ intersect $U$
in a Baer subline, is an ovoidal Buekenhout-Metz unital (where $\epsilon\approx
2q$ for $q$ even and $\epsilon\approx q^{3/2}/2$ for $q$ odd).
