  We address the nonperturbative calculation of the inclusive decay rate of
semileptonic $B_{(s)}$-meson decays from lattice QCD. Precise Standard-Model
predictions are key ingredients in searches for new physics, and this type of
computation may eventually provide new insight into the long-standing tension
between the inclusive and exclusive determinations of the
Cabibbo-Kobayashi-Maskawa (CKM) matrix elements $|V_{cb}|$ and $|V_{ub}|$. We
present results from a pilot lattice computation for $B_s \rightarrow X_c\, l
\nu_l$, where the initial $b$ quark described by the relativistic-heavy-quark
(RHQ) formalism on the lattice and the other valence quarks discretised with
domain-wall fermions are simulated approximately at their physical quark
masses. We compare two different methods for computing the decay rate from
lattice data of Euclidean $n$-point functions, namely Chebyshev and
Backus-Gilbert approaches. We further study how much the ground-state meson
dominates the inclusive decay rate and indicate our strategy towards a
computation with a more comprehensive systematic error budget.
