  We compute the electrical conductivity of the strongly interacting medium in
the presence of strong magnetic background fields, $eB=4,9~GeV^2$, and for
different values of the temperature, both in the confined and in the deconfined
Quark-Gluon Plasma (QGP) phase. The conductivity is obtained from the Euclidean
lattice time correlator of the electrical current, computed on gauge
configurations sampled from Monte-Carlo simulations of an improved staggered
discretization of $N_f = 2+1$ QCD. We perform the inverse Laplace transform of
the correlator adopting a recently-proposed version of the standard
Backus--Gilbert procedure for the inversion. The results obtained in the QGP
phase show a sizable enhancement of the conductivity in the direction parallel
to the magnetic field, as well as a suppression in the direction orthogonal to
it. Such enhancement could be attributed to the manifestation of the Chiral
Magnetic Effect (CME): following this guess, we extract the behaviour of the
relaxation time of this process, extrapolate it to the continuum limit and
compare it to previous results, finding it lower than expected in the explored
range of temperatures.
