Best uniform rational approximation of $|x|$ on $[-1,1]$

We consider best rational approximants in the uniform norm to the function $|x|$ on $[-1,1]$. The main result is a proof of a conjecture by R. S. Varga, A. Ruttan, and A. J. Carpenter. They have conjectured that if $E_{nn}(|x|,[-1,1])$, $n\in\mathbb{N}$, denotes the error of the $n$th degree rational approximant, then
\begin{equation} \lim_{n\to\infty}e^{\pi\sqrt n}E_{nn}(|x|,[-1,1])=8. \tag{1} \end{equation}
This conjecture generalizes earlier results, among them most prominently results by D. J. Newman and by N. S. Vyacheslavov.