  Measuring vector-boson scattering beyond the fully-leptonic final state is
becoming possible at the LHC, which demands to have a solid control on the
theory predictions for all final states of this class of processes. In this
work we present a full off-shell leading-order calculation for the process
$\mathrm{p}\mathrm{p} \to \ell \nu_\ell + 4\mathrm{j}$ in two fiducial regions
which are particularly relevant for its experimental measurement. In addition
to the fully electroweak order, i.e. $\mathcal{O}(\alpha^6)$, we complement our
results with $\mathcal{O}(\alpha_\mathrm{s}\alpha^5)$ and
$\mathcal{O}(\alpha_\mathrm{s}^2\alpha^4)$ for inclusive predictions. At
$\mathcal{O}(\alpha^6)$ we present for the first time a systematic treatment of
the process in double-pole approximation and we perform a detailed study of its
range of validity by considering inclusive and differential predictions
compared to the full off-shell calculation.
