Generalized Hovey conjecture for p-uniform hypertrees

A $p$-uniform hypertree is a uniform hypergraph in which every hyperedge has $p$ vertices and whose underlying incidence structure is a hypertree. A $k$-cordial labeling assigns labels from $\mathbb{Z}_k$ to the vertices, with the vertex and induced hyperedge labels distributed as evenly as possible.

Generalized Hovey conjecture. All $p$-uniform hypertrees are $k$-cordial for all $k$.

This is proposed as the main open problem generalizing Hovey's conjecture from trees to uniform hypertrees. The paper establishes the ordinary cordial case, but leaves the assertion for every $k$ open.