Multicolor Ramsey conjecture for disjoint unions of three-vertex paths

Let $P_3$ be the path on three vertices, and let $gr_k(K_3:n_1P_3,\ldots,n_kP_3)$ denote the corresponding Gallai-Ramsey number. For positive integers $k,n_1,n_2,\ldots,n_k$ with $n_1=\max\{n_i\}$, the multicolor $P_3$ Ramsey conjecture.

$$R(n_1P_3,n_2P_3,\ldots,n_kP_3)=gr_k(K_3:n_1P_3,n_2P_3,\ldots,n_kP_3)=2n_1+1+\sum_{i=1}^k(n_i-1).$$

The claim would determine the general multicolor classical Ramsey number for these linear forests and identify it with the Gallai-Ramsey number; the source says that this classical number is not known in general.