  We investigate inflation driven by the Higgs boson in the Palatini
formulation of General Relativity. Our analysis primarily focuses on a small
non-minimal coupling of the Higgs field to gravity in the range $0<\xi\lesssim
1$. We incorporate the renormalization group running of the relevant parameters
as computed within the Standard Model and allow for small corrections. In
addition to $\xi$, our model features two tunable parameters: the low-energy
value of the top Yukawa coupling and an effective jump of the Higgs
self-interaction. Our results indicate that critical points leading to a large
enhancement of the power spectrum can be produced. However, the observed
amplitude of perturbations in the CMB cannot be matched within this setting. On
the one hand, this makes it difficult to generate a sizable abundance of
primordial black holes. On the other hand, our finding can be viewed as further
evidence that Palatini Higgs inflation has favourable high-energy properties
due to robustness against quantum corrections.
