Semistable conjecture via K-theory

We show that the semistable conjecture of Fontaine and Jannsen (see [9]) is true for proper, vertical, fine, and saturated log-smooth families with reduction of Cartier type (e.g., proper schemes with simple semistable reduction). We derive it from Suslin's comparison theorem [31, Corollary 4.3] between motivic cohomology and étale cohomology. This gives a new proof of the semistable conjecture showing motivic character of p-adic period maps