Does the sequence $\sin(n!\pi^2)$ converge or diverge?

Does the sequence $\sin(n!\pi^2)$ converge or diverge?

I devoted a chapter (3) of my thesis on the set $\displaystyle G = \{x \in \mathbb{R}: \lim_{n \rightarrow \infty} \sin{(n! \pi x) = 0}\}$. It is easy to see that (Euler's constant) $e \in G$. Let alone $\pi$, I do not even know if $e^2 \in G$.

I cant answer the question but here are some thoughts. $\sin(n \pi^2)$ does not have a limit because $\{ n\pi \}$ is dense in $[0,1]$. So the question would really be is $\{ n! \pi \}$ is dense. One might ask if $\{ n! \alpha \}$ is dense for any irrational $\alpha$, but it is easy to see that $\{ n! e \}\rightarrow 0$ from the series definition of $e$. So $\sin(n! e \pi)$ limits to $0$. Thus the behaviour of $\{ n! \pi \}$ depends on properties of $\pi$.