  We study reionization in two non-flat $\Lambda$CDM inflation models that best
fit the Planck 2015 cosmic microwave background anisotropy observations,
ignoring or in conjunction with baryon acoustic oscillation distance
measurements. We implement a principal component analysis (PCA) to estimate the
uncertainties in the reionization history from a joint quasar-CMB dataset. A
thorough Markov Chain Monte Carlo analysis is done over the parameter space of
PCA modes for both non-flat $\Lambda$CDM inflation models as well as the
original Planck 2016 tilted, spatially-flat $\Lambda$CDM inflation model.
Although both flat and non-flat models can closely match the low-redshift
($z\lesssim6$) observations, we notice a possible tension between high-redshift
($z\sim8$) Lyman-$\alpha$ emitter data and the non-flat models. This is solely
due to the fact that the closed models have a relatively higher reionization
optical depth compared to the flat one, which in turn demands more
high-redshift ionizing sources and favors an extended reionization starting as
early as $z\approx14$. We conclude that as opposed to flat-cosmology, for the
non-flat cosmology models (i) the escape fraction needs steep redshift
evolution and even unrealistically high values at some redshifts and (ii) most
of the physical parameters require to have non-monotonic redshift evolution,
especially apparent when Lyman-$\alpha$ emitter data is included in the
analysis.
