We study the amplitude-constrained additive white Gaussian noise channel. It is well known that the capacity-achieving input distribution for this channel is discrete and supported on finitely many points. The best known bounds show that the support size of the capacity-achieving distribution is lower-bounded by a term of order $A$ and upper-bounded by a term of order $A^2$, where $A$ denotes the amplitude constraint. It was conjectured in [1] that the linear scaling is optimal. In this work, we establish a new lower bound of order $A\sqrt{\log A}$, improving the known bound and ruling out the conjectured linear scaling. To obtain this result, we quantify the fact that the capacity-achieving output distribution is close to the uniform distribution in the interior of the amplitude constraint. Next, we introduce a wrapping operation that maps the problem to a compact domain and develop a theory of best approximation of the uniform distribution by finite Gaussian mixtures. These approximation bounds are then combined with stability properties of capacity-achieving distributions to yield the final support-size lower bound.