Unitals with many Baer secants through a fixed point

We show that a unital $U$ in $\mathrm{PG}(2,q^2)$ 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 $ε\approx 2q$ for $q$ even and $ε\approx q^{3/2}/2$ for $q$ odd).

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