Irreducible smooth representations in defining characteristic without central character

Let $p>3$ be a prime, $n>1$ be an integer, and $F$ be a non-archimedean local field with residue field a proper finite extension of $\mathbb{F}_p$. Let $E$ be an algebraically closed countable field extension of the residue field of $F$. In this short note, we explain how the methods from arXiv:1809.10247 and arXiv:2210.07281 can be used to construct irreducible smooth representations of $\mathrm{GL}_n(F)$ over $E$ without a central character. We also construct irreducible smooth representations of $\mathrm{GL}_n(F)$ over $E$ with simultaneously a central character, nonscalar endomorphisms, and if $n>3$, without a Hecke eigenvalue.