Sublines of prime order contained in the set of internal points of a conic

In \cite{BALLBLOKHUISLAVRAUW200*} it was shown that if $q \geq 4n^2-8n+2$ then there are no subplanes of order $q$ contained in the set of internal points of a conic in $\PG(2,q^n)$, $q$ odd, $n\geq 3$. In this article we improve this bound in the case where $q$ is prime to $q > 2n^2-(4-2\sqrt{3})n+(3-2\sqrt{3})$, and prove a stronger theorem by considering sublines instead of subplanes. We also explain how one can apply this result to flocks of a quadratic cone in $\PG(3,q^n)$, ovoids of $Q(4,q^n)$, rank two commutative semifields, and eggs in $\PG(4n-1,q)$.