The Gallai-Ramsey conjecture for even cycles and paths

Let $n\ge3$ and $k\ge2$. For each $i\in\{0,1,\ldots,n-2\}$, let $G_i=P_{2i+3}$, and let $G_{n-1}$ be either $C_{2n}$ or $P_{2n+1}$. Choose integers $i_1,\ldots,i_k$ satisfying

$$n-1\ge i_1\ge\cdots\ge i_k\ge0.$$

Here $GR(G_{i_1},\ldots,G_{i_k})$ is the least order of a complete graph such that every Gallai edge-coloring using at most $k$ colors contains a monochromatic copy of $G_{i_j}$ in color $j$ for some $j$. The Gallai-Ramsey conjecture for even cycles and paths. For all such $n$, $k$, and indices,

$$GR(G_{i_1},\ldots,G_{i_k})=|G_{i_1}|+\sum_{j=2}^k i_j.$$

The displayed quantity is known to be a lower bound by an explicit Gallai-coloring construction; the conjecture asserts that this lower bound is always sharp. The source gives no resolution of the general upper bound.