  It was conjectured, independently by two sets of authors, that for all
integers $k,d\ge 1$ there exists $\ell>0$, such that if $S,T$ are subsets of
vertices of a graph $G$, then either there are $k$ paths between $S,T$,
pairwise at distance at least $d$, or there is a set $X\subseteq V(G)$ with
$|X|\le k-1$ such that every path between $S,T$ contains a vertex with distance
at most $\ell$ from some member of $X$. The result is known for $k\le 2$, but
we will show that it is false for all $k\ge 3$, even if $G$ is constrained to
have maximum degree at most three. We also give a proof of the result when
$k=2$ that is simpler than the previous proofs.
