  A graph $G$ is terminal-pairable with respect to a demand multigraph $D$ on
the same vertex set as $G$, if there exists edge-disjoint paths joining the end
vertices of every demand edge of $D$. In this short note, we improve the upper
bound on the largest $\Delta(n)$ with the property that the complete graph on
$n$ vertices is terminal-pairable with respect to any demand multigraph of
maximum degree at most $\Delta(n)$. This disproves a conjecture originally
stated by Csaba, Faudree, Gy\'arf\'as, Lehel and Schelp.
