  The inclusive hadronic decay-rates of the $\tau$ lepton are particularly
interesting from the phenomenological point of view since they give access to
the CKM matrix elements $V_{ud}$ and $V_{us}$. In this talk, we discuss how a
recent method for the extraction of smeared spectral densities from Euclidean
lattice correlators can be used to obtain a direct lattice determination of
inclusive hadronic $\tau$ decay rates. We also present preliminary numerical
results obtained by applying this method to correlators measured on two gauge
ensembles produced by the ETMC with $N_f=2+1+1$ dynamical flavours at physical
pion masses, lattice spacing $a\simeq 0.08$ fm and volumes $L\simeq 5.1$ fm and
$L\simeq 7.6$ fm.
