Large sum-free sets in finite vector spaces II

Answering a question of Leo Versteegen, we prove that for $n\ge 3$ every sum-free set $A\subseteq {\mathbb{F}}_5^n$ with $\mathrm{\mid }A\mathrm{\mid }\ge 28\cdot 5^{n-3}$ is either contained in the union of two parallel hyperplanes, or isomorphic to $\mathrm{\Lambda }\times {\mathbb{F}}_5^{n-3}$, where $\mathrm{\Lambda }\subseteq {\mathbb{F}}_5^3$ denotes a certain sum-free set of size $28$ discovered by Vsevolod Lev and Leo Versteegen.