Cohen-Macaulayness of absolute integral closures

We prove that, modulo any power of a prime p, the absolute integral closure of an excellent noetherian domain is Cohen-Macaulay. A graded analog is also established, yielding variants of Kodaira vanishing "up to finite covers" in mixed characteristic. Our main tools are (log) prismatic cohomology (which yields a Frobenius action in mixed characteristic) and the p-adic Riemann-Hilbert functor for constructible étale Fₚ-sheaves on varieties over a p-adic field (which almost controls perfectified prismatic cohomology).