KK-theory of log-schemes II: Log-syntomic KK-theory

We prove that suitably truncated topological log-syntomic-étale homotopy KK-theory of proper semistable schemes in mixed characteristic surjects onto the Kummer log-étale pp-adic KK-theory of the generic fiber. As a corollary we get that the log-syntomic-étale cohomology is a direct factor of the log-syntomic-étale cohomology of homotopy KK-theory sheaves. The proofs use pp-adic Hodge theory computations of pp-adic nearby cycles.