The cohomological index of free $\mathbb{Z}\mathrm{/}p$-actions is not additive with respect to join

Let $p$ denote an odd prime. We show by example that the inequalities obtained in arXiv:1704.05827 for the behaviour of the cohomological index of a join of free $\mathbb{Z}\mathrm{/}p$-actions are sharp. Namely, for all odd integers $k,l$ at least one of which is greater than one, we give examples of finite free $\mathbb{Z}\mathrm{/}p$-CW complexes of cohomological indices $k$ and $l$ whose join has index $k+l+1$ and also examples where the join has index $k+l-1$.