Gallai-Ramsey formula for disjoint unions of paths on three vertices

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

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

This extends the known two-color Ramsey formula for copies of $P_3$ to general Gallai colorings, but the source does not report a resolution.