On the smallest non-trivial tight sets in Hermitian polar spaces H(d,q2), d even

We show that every $x$-tight set of a Hermitian polar spaces $\mathrm{H}(2n,q^2)$, $n\ge 2$, is the union of $x$ disjoint generators of the polar space provided that $x\le \frac{1}{2}(q+1)$. This was known before only when $n\in \{2,3\}$. This result is a contribution to the conjecture that the smallest $x$-tight set of $H(2n,q^2)$ that is not a union of disjoint generators occurs for $x=q+1$ and is for sufficiently large $q$ an embedded subgeometry.