Let $p$ be a prime number. Let $X\to\operatorname{Spec}\mathbb Z_p$ be smooth and proper. Is it true that the map $H^i(X,\Omega^{\bullet\geq j}_{X/\mathbb Z_p})\to H^i(X,\Omega^\bullet_{X/\mathbb Z_p})$ induced by the inclusion of complexes is injective? This is easy to see if the groups $H^i(X,\Omega^j_{X/\mathbb Z_p})$ are $p$-torsion free, but I think it should be true in general.

Thanks!