How Many Modes Can a Mixture of Gaussians with Uniformly Bounded Means Have?

We show, by an explicit construction, that a mixture of univariate Gaussian densities with variance $1$ and means in $[-A,A]$ can have $\mathrm{\Omega }(A^2)$ modes. This disproves a recent conjecture of Dytso, Yagli, Poor and Shamai \cite{DYPS20} 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 $\mathrm{\Omega }(A^{2d})$ modes.