How many modes can a constrained Gaussian mixture have

We show, by an explicit construction, that a mixture of univariate Gaussians with variance 1 and means in $[-A,A]$ can have $\Omega(A^2)$ modes. This disproves a recent conjecture of Dytso, Yagli, Poor and Shamai [IEEE Trans. Inform. Theory, Apr. 2020], who showed that such a mixture can have at most $O(A^2)$ modes and surmised that the upper bound could be improved to $O(A)$. Our result holds even if an additional variance constraint is imposed on the mixing distribution. Extending the result to higher dimensions, we exhibit a mixture of Gaussians in $\mathbb{R}^d$, with identity covariances and means inside $[-A,A]^d$, that has $\Omega(A^{2d})$ modes.