Packing $T$-connectors in graphs needs more connectivity

Strengthening the classical concept of Steiner trees, West and Wu [J. Combin. Theory Ser. B 102 (2012), 186–205] introduced the notion of a $T$-connector in a graph $G$ with a set $T$ of terminals. They conjectured that if the set $T$ is $3k$-edge-connected in $G$, then $G$ contains $k$ edge-disjoint $T$-connectors. We disprove this conjecture by constructing infinitely many counterexamples for $k=1$ and for each even $k$.