Hermite-Biehler functions with zeros close to the imaginary axis

A Hermite-Biehler function $E$ gives rise to a de Branges Hilbert space $\mathcal{H}(E)$ consisting of entire functions. We are going to show that for Hermite-Biehler functions of sufficiently small growth and a certain distribution of zeros every proper de Branges subspace of $\mathcal{H}(E)$ coincides for some $n\in\mathbb{N}$ with the $(n+1)$-dimensional linear space of all polynomials of degree at most $n$.