  We present the lattice QCD calculation of inclusive semileptonic $B_s$-meson
decays. We follow a recently proposed method, which is based on the extraction
of smeared spectral densities from Euclidean correlation functions and on the
numerical reconstruction of the integration kernel relevant for the inclusive
decay rate calculation. We compute four-point Euclidean correlation functions
using JLQCD and ETM gauge ensembles with unphysically light $b$-quark masses,
and apply two different methods for the integration kernel reconstruction.
Finally, we show that the lattice results obtained in this work are in good
agreement with the analytic predictions of the operator-product-expansion. This
opens the path for a future full lattice QCD calculation to be used as
theoretical input for the determination of the magnitude of the CKM element
$V_{cb}$.
