The asymptotic of off-diagonal online Ramsey numbers for paths

We prove that for every $k\ge 10$, the online Ramsey number for paths $P_k$ and $P_n$ satisfies $\tilde{r}(P_k,P_n)\ge \frac{5}{3}n+\frac{k}{9}-4$, matching up to a linear term in $k$ the upper bound recently obtained by Bednarska-Bzd{ę}ga. In particular, this implies ${\lim }_{n\to \mathrm{\infty }}\frac{\tilde{r}(P_k,P_n)}{n}=\frac{5}{3}$, whenever $10\le k=o(n)$, disproving a conjecture by Cyman, Dzido, Lapinskas and Lo.