Cordial labeling conjecture for hypertrees

Let $H=(V,E)$ be a hypertree, meaning a connected hypergraph without cycles. For an integer $k\ge 2$, a vertex labeling $f:V\to\mathbb{Z}_k$ induces an edge labeling by $f(e)=\sum_{v\in e}f(v)$ modulo $k$. The hypergraph is $k$-cordial when the numbers of vertices and edges receiving any two labels in $\mathbb{Z}_k$ differ by at most $1$. Hypertree cordiality conjecture. Every hypertree is $k$-cordial for every $k\ge 2$. The paper proves the cases $k\in\{2,3\}$, while the proposed assertion for all larger values remains open and is presented as a generalization of Hovey's conjecture.