  We prove the main conjecture of arXiv:2102.13459 for integral coefficients in
the case of tori and prove that it is t-exact. Along the way we prove that the
spectral action as constructed in that manuscript is compatible with the action
of the excursion algebra and preserves the grading by $\pi_1(G)_Q$ on both
sides. We additionally develop a (non-solidified) version of condensed group
(co)homology and show that many constructions from classical group (co)homology
extend to that case.
