  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.
