Unitals with many Baer secants through a fixed point

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