Caro's zero-sum Ramsey conjecture for trees modulo 3

Let $T$ be a tree on $n \equiv 1 \pmod{3}$ vertices. Caro's conjecture.

$$R(T,\mathbb{Z}_3) = \begin{cases} n+2, & \text{if } T \text{ is the star } K_{1,n-1} \\ n+1, & \text{if } T \text{ has no vertex of degree } 0 \pmod{3} \\ n, & \text{otherwise.} \end{cases}$$

This conjecture extends the understanding of zero-sum Ramsey numbers from $\mathbb{Z}_2$ to the next cyclic group, $\mathbb{Z}_3$. The source presents it as proposed by Caro in 2019; no resolution is given here.