A log prismatic-crystalline comparison theorem

We show a comparison theorem between log prismatic cohomology and log crystalline cohomology for a $p$-adic formal scheme with semistable reduction. Combined with the prismatic-étale comparison theorem recently proved by Tian, this implies the $C_{\mathrm{s}\mathrm{t}}$-conjecture in the semistable case with coefficients given by semistable local systems.