On the $p$-adic local invariant cycle theorem

For a proper, flat, generically smooth scheme $X$ over a complete DVR with finite residue field of characteristic $p$, we define a specialization morphism from the rigid cohomology of the geometric special fibre to $D_{crys}$ of the $p$-adic \'etale cohomology of the geometric generic fibre, and we make a conjecture ("$p$-adic local invariant cycle theorem") that describes the behavior of this map for regular $X$, analogous to the situation in $l$-adic \'etale cohomology for $l\neq p$. Our main result is that, if $X$ has semistable reduction, this specialization map induces an isomorphism on the slope $[0,1)$-part.