  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 \'etale $\mathbf{F}_p$-sheaves on
varieties over a $p$-adic field (which almost controls perfectified prismatic
cohomology).
