Purity conjecture for zero-sumfree complexes at half the modulus

For an odd integer $n$, consider the complexes $\Delta_{n,(n-1)/2}$ and $\Delta_{n,(n+1)/2}$ of $\ell$-zero-sumfree subsets. A simplicial complex is pure when all of its facets have the same dimension. Purity conjecture. Each of these two complexes is pure if and only if $n$ is prime. The conjecture was generated from computations for $n\leq19$ and concerns the relationship between purity and primality; the source gives no resolution status.