Hypergraph Nash–Williams–Tutte conjecture in literal form

A tree assignment of a hypergraph $H$ chooses, for each hyperedge occurrence $e$, a tree on vertex set $e$, with every edge of that tree labelled by $e$. A $k$-tree decomposition of the resulting labelled graph is an ordered partition of its edges into $k$ spanning trees. Such a graph is $k$-distinguishable if it has a $k$-tree decomposition uniquely determined by its signature, where the signature records, for each hyperedge occurrence, the weighted number of its assigned tree edges in each part. A hypergraph is $k$-weakly-partition-connected when it satisfies the corresponding weak partition-connectivity condition, and write $t$ for its number of vertices.

Hypergraph Nash–Williams–Tutte conjecture. For positive integers $t$ and $k$, every $k$-weakly-partition-connected hypergraph $H$ on $t$ vertices has a $k$-distinguishable tree assignment.

The paper states that this literal formulation is false: an excess obstruction shows that any $k$-distinguishable tree assignment would require the equality $\rho(H)=k(t-1)$, while weak partition connectivity permits hypergraphs with $\rho(H)>k(t-1)$. Thus overfull weakly partition-connected hypergraphs provide counterexamples.