Numerical results on best uniform rational approximation of $|x|$ on $[-1,1]$

With $E_{n,n}(|x|;[-1,1])$ denoting the error of best uniform rational approximation from $\pi_{n,n}$ to $|x|$ on $[-1,1]$, we determine the numbers $\{E_{2n,2n}(|x|;[-1,1])\}_{n=1}^{40}$, where each of these numbers was calculated with a precision of at least 200 significant digits. With these numbers, the Richardson extrapolation method was applied to the products $\{e^{\pi\sqrt{2n}}E_{2n,2n}(|x|;[-1,1])\}_{n=1}^{40}$, and it appears, to at least 10 significant digits, that
$$ 8\stackrel{?}{=}\lim_{n\to\infty}e^{\pi\sqrt{2n}}E_{2n,2n}(|x|;[-1,1]), $$
which gives rise to an interesting new conjecture in the theory of rational approximation.