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 $\tilde{r}(P_k,P_n) \geq \frac{5}{3}n + \frac{k}{9} - 4$, matching up to a linear term in $k$ the upper bound recently obtained by Bednarska-Bzd{\k{e}}ga. In particular, this implies $\lim_{n \rightarrow \infty} \frac{\tilde{r}(P_k, P_n)}{n} = \frac{5}{3}$, whenever $10 \le k=o(n)$, disproving a conjecture by Cyman, Dzido, Lapinskas and Lo.