Determine the multicolour Ramsey number $R_k(C_{2n})$ of even cycles (Erdős #555)

For a graph $G$, let $R_k(G)$ denote the $k$-colour Ramsey number: the least $m$ such that every colouring of the edges of $K_m$ with $k$ colours contains a monochromatic copy of $G$. Determine the value (or the sharp asymptotics in $k$) of $$R_k(C_{2n}),$$ where $C_{2n}$ is the even cycle on $2n$ vertices, for each fixed $n\geq 2$.

Acceptance. FULLY RESOLVES: a proof determining $R_k(C_{2n})$ exactly (a closed form valid for all $k$) for some fixed $n\geq 2$, OR determining the sharp leading-order asymptotics — i.e. closing the exponent gap between $k^{1+1/(2n)}$ and $k^{1+1/(n-1)}$ (in particular fixing the constant for $R_k(C_4)=(c+o(1))k^2$) — with proof. ADVANCES: compute new exact values of $R_k(C_4)$ (or $R_k(C_{2n})$) beyond those recorded in OEIS A389313, each certified by a colouring witness attaining the lower bound and an exhaustiveness certificate for the upper bound; OR improve the best-known bounds stated in the background (the $k^{1\pm}$ exponents, or the Chung–Graham additive constants for $C_4$), with proof. Deliver the proof, or the witness colourings plus exhaustiveness certificates plus an extended value table.