Non-admissibility of some universal supersingular representations

Let $K\mathrm{/}{\mathbf{Q}}_p$ be an unramified extension of degree $f$ with residue field $k$. Let $\sigma$ be an irreducible representation of ${\mathrm{G}\mathrm{L}}_n(k)$ over ${\bar{\mathbf{F}}}_p$. For $n\ge 3$, we prove that the universal supersingular representation of weight $\sigma$ is non-admissible and of infinite length when $\sigma$ is sufficiently generic and satisfies certain technical conditions. This generalizes the previous results for $n=2$ and a non-trivial finite extension $K\mathrm{/}{\mathbf{Q}}_p$. Our method employs a weight cycling argument together with recent progress on the Serre weight conjectures.