A problem of Beelen, Garcia and Stichtenoth on an Artin-Schreier tower in characteristic two

We study a tower of function fields of Artin-Schreier type over a finite field with $2^s$ elements. The study of the asymptotic behavior of this tower was left as an open problem by Beelen, Garc\'ia and Stichtenoth in $2006$. We prove that this tower is asymptotically good for $s$ even and asymptotically bad for $s$ odd.