  We prove equality of the various $p$-adic period morphisms for smooth, not
necessarily proper, schemes. We start with showing that the $K$-theoretical
uniqueness criterium we had found for proper smooth schemes extends to proper
finite simplicial schemes in the good reduction case and to cohomology with
compact support in the semistable reduction case. It yields the equality of the
period morphisms for cohomology with compact support defined using the
syntomic, almost \'etale, and motivic constructions. We continue with showing
that the $h$-cohomology period morphism agrees with the syntomic and almost
\'etale period morphisms whenever the latter morphisms are defined. We do it by
lifting the syntomic and almost \'etale period morphisms to Voyevodsky
triangulated category of motives, where their equality with the $h$-cohomology
period morphism can be checked directly using the Beilinson Poincar\'e Lemma.
