  We report a study of the spin dynamics of the frustrated ferromagnetic
$J_1$-$J_2$ chain compound linarite, PbCuSO$_4$(OH)$_2$, in applied magnetic
fields up to field polarization. By means of an extreme-environment inelastic
neutron scattering experiment, we have measured the low-energy spin excitations
of linarite in fields up to 8.8 T for $\mathbf{H} \parallel b$ axis. We have
recorded the spin excitation spectra along $h$, $k$ and $l$ for the
field-induced magnetic phases IV, V and the field polarized state close to
saturation. By employing first-principles calculations, we estimate the leading
magnetic exchanges out of the $bc$ plane and model the dispersion relations
using linear spin-wave theory. In this way, we find evidence for a (very weak)
residual magnetic exchange coupling out of the $bc$ plane. Together with the
previously established dominant intrachain couplings $J_1$ and $J_2$ and the
interchain coupling $J_3$, we derive an effective set of exchange couplings for
a microscopic description of linarite. Further, we find that the peculiar
character of phase V manifests itself in the measured spin dynamics.
