Asymptotic online size Ramsey conjecture for even cycles versus paths

Let $C_{2k}$ be the cycle of length $2k$, let $P_n$ be the path on $n$ vertices, and let $\tilde{r}(G,H)$ denote the online size Ramsey number for graphs $G$ and $H$. The parameter $k$ is fixed.

Even-cycle online Ramsey conjecture.

$$\tilde{r}(C_{2k},P_n)=2n+o(n)\quad\text{for every fixed }k\ge 3.$$

The theorem in the paper establishes the exact value $\tilde{r}(C_4,P_n)=2n-2$ for $n\ge 8$, while previously known bounds show that avoiding odd cycles is asymptotically more difficult for Painter. The conjecture asserts that every fixed even cycle of length at least six has the same leading asymptotic online size Ramsey number as $C_4$ against a path.