Caro–Roditty conjecture for zero-sum tree embeddings

Prove that for integers m≥k≥2 with k dividing m, every graph G with minimum degree at least n+k−2 forces, under every Z_k-edge-colouring, a zero-sum modulo k copy of every tree T_n on n=m+1 vertices.

Background

This conjecture proposes a minimum-degree condition guaranteeing zero-sum copies of trees under group-valued edge-colourings. If true, it would imply the upper bound R(T_n,Z_k)≤n+k−1, and in particular R(T_n,Z_3)≤n+2. The paper uses this implication as motivation for its results but does not prove the conjecture in full generality.