Constraints on reionisation history obtained with Planck (in grey, Planck Collaboration I 2020) and SPT high-ℓ data (in blue, Reichardt et al. 2021) for cosmology-dependent kSZ and tSZ spectra. Cosmology is free but, for the latter, Gaussian priors from Sect. 4.1 are imposed on four cosmological parameters. Inferences are compared to currently available data points on the ionisation fraction of the IGM from Lyman-α emitters and QSO spectra (see text for references).