Does $\pi$ satisfy the law of the iterated logarithm?

Does $\pi$ satisfy the law of the iterated logarithm?

It is widely conjectured that $\pi$ is normal in base $2$.
But what about the law of the iterated logarithm?
Namely, if $x_n$ is the $n$th binary digit of $\pi$, does it seem likely
(from computer experiments for example) that the following holds?
$$\limsup_{n\rightarrow\infty} \frac{S_n }{\sqrt{n\log\log
n}}=\sqrt{2}\quad\text{where}\quad S_n=2(x_1 + \ldots + x_n) - n$$
What about other (conjectured) normal numbers like $e$ and $\sqrt{2}$?



I am sorry if this is too easy, but I tried to search for it and I could
not find in on the Internet. I suppose I could run an experiment myself,
but I assumed this is well known, and I would need to brush up on my
programming skills to do so...
Also, I am sorry that I really don't know how to properly tag this.