Centrality of Gaussian distribution for LWE error

Consider the LWE problem.

Let $A$ be an $m \times n$ matrix, $x$ is an $n \times 1$ vector, $u$ is a $m \times 1$ vector, and $e$ is sampled from a Gaussian distribution.

We are given either $Ax + e ~~(mod~q)$ or $u ~(mod~q)$ the conjecture being that it is difficult to distinguish between these samples in polynomial time, with high probability over the choice of $A$, $x$, $u$ and $e$ (for appropriate choices of $m$ and $q$.)

I wanted to ask about the centrality of the Gaussian distribution while considering the security of LWE.

Is LWE hard if $e$ is sampled from other distributions — like the uniform distribution or the exponential distribution?