  Let $n>m\geqslant 1$ be integers with $n+m\geqslant 4$ even. We prove the
existence of Maass forms with large sup norms on anisotropic ${\rm O}(n,m)$, by
combining a counting argument with a new period relation showing that a certain
orthogonal period on ${\rm O}(n,m)$ distinguishes theta lifts from ${\rm
Sp}_{2m}$. This generalizes a method of Rudnick and Sarnak in the rank one
case, when $m = 1$. Our lower bound is naturally expressed as a ratio of the
Plancherel measures for the groups ${\rm O}(n,m)$ and ${\rm
Sp}_{2m}(\mathbb{R})$, up to logarithmic factors, and strengthens the lower
bounds of our previous paper for such groups. In the case of odd-dimensional
hyperbolic spaces, the growth exponent we obtain improves on a result of
Donnelly, and is optimal under the purity conjecture of Sarnak.
