  A cubical polytope is a polytope with all its facets being combinatorially
equivalent to cubes. The paper is concerned with the linkedness of the graphs
of cubical polytopes.
  A graph with at least $2k$ vertices is \textit{$k$-linked} if, for every set
of $k$ disjoint pairs of vertices, there are $k$ vertex-disjoint paths joining
the vertices in the pairs. We say that a polytope is \textit{$k$-linked} if its
graph is $k$-linked.
  In a previous paper \cite{BuiPinUgo20a} we proved that every cubical
$d$-polytope is $\floor{d/2}$-linked. Here we strengthen this result by
establishing the $\floor{(d+1)/2}$-linkedness of cubical $d$-polytopes, for
every $d\ne 3$.
  A graph $G$ is {\it strongly $k$-linked} if it has at least $2k+1$ vertices
and, for every vertex $v$ of $G$, the subgraph $G-v$ is $k$-linked.
  We say that a polytope is (strongly) \textit{$k$-linked} if its graph is
(strongly) $k$-linked. In this paper, we also prove that every cubical
$d$-polytope is strongly $\floor{d/2}$-linked, for every $d\ne 3$.
  These results are best possible for this class of polytopes.
