Caro–Roditty conjecture on zero-sum tree embeddings

Let $m \geq k \geq 2$ be integers such that $k \mid m$, and let $T_{m+1}$ be a tree on $n=m+1$ vertices. Let $G$ be a graph with minimum degree at least $n+k-2$. Caro–Roditty conjecture. Every $\mathbb{Z}_k$-edge-colouring of $G$ forces a zero-sum modulo $k$ copy of $T_n$.

The conjecture would give upper bounds for zero-sum Ramsey numbers of trees, including $R(T_n,\mathbb{Z}_k)\leq n+k-1$; the source explicitly describes it as open. Its implication for the paper's case is the bound $R(T_n,\mathbb{Z}_3)\leq n+2$.