Caro’s conjectured classification of zero-sum Ramsey numbers for trees

Prove that for every tree T on n≡1 modulo 3 vertices, R(T,Z_3) equals n+2 when T is the star K_{1,n−1}, equals n+1 when T has no vertex whose degree is congruent to 0 modulo 3, and equals n otherwise.

Background

The conjecture gives a complete classification of zero-sum Ramsey numbers modulo 3 for trees whose order is congruent to 1 modulo 3. The paper proves the conjectured values for several infinite families, but explicitly treats the full classification as unresolved and returns to it in the concluding remarks.