  We show that a mapping $f\colon X\to Y$ satisfying the metric condition of
quasiconformality outside suitable exceptional sets is in the Newton-Sobolev
class $N_{\textrm{loc}}^{1,1}(X;Y)$. Contrary to previous works, we only assume
an asymptotic version of Ahlfors-regularity on $X,Y$. This allows many
non-Ahlfors regular spaces, such as weighted spaces and Fred Gehring's bowtie,
to be included in the theory. Unexpectedly, already in the classical setting of
unweighted Euclidean spaces, our theory detects Sobolev mappings that are not
recognized by previous results.
