  We measure the intergalactic medium (IGM) opacity in the Ly$\alpha$ as well
as in the Ly$\beta$ forest along $19$ quasar sightlines between $5.5\lesssim
z_{\rm abs}\lesssim 6.1$, probing the end stages of the reionization epoch.
Owing to its lower oscillator strength the Ly$\beta$ transition is sensitive to
different gas temperatures and densities than Ly$\alpha$, providing additional
constraints on the ionization and thermal state of the IGM. A comparison of our
measurements to different inhomogeneous reionization models, derived from
post-processing the Nyx cosmological hydrodynamical simulation to include
spatial fluctuations in the ultraviolet background (UVB) or the gas temperature
field, as well as to a uniform reionization model with varying thermal states
of the IGM, leads to two primary conclusions: First, we find that including the
effects of spectral noise is key for a proper data to model comparison. Noise
effectively reduces the sensitivity to high opacity regions, and thus even
stronger spatial inhomogeneities are required to match the observed scatter in
the observations than previously inferred. Second, we find that models which
come close to reproducing the distribution of Ly$\alpha$ effective optical
depths nevertheless underpredict the Ly$\beta$ opacity at the same spatial
locations. The origin of this disagreement is not entirely clear but models
with an inversion in the temperature-density relation of the IGM just after
reionization is completed match our measurements best, although they still do
not fully capture the observations at $z\gtrsim 5.8$.
