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