Cyman–Dzido–Lapinskas–Lo asymptotic conjecture for online Ramsey numbers of paths

For a positive integer $n$, let $P_n$ denote the path consisting of $n$ vertices, and let $\tilde{r}(P_k,P_n)$ be the online Ramsey number for the pair of paths $P_k$ and $P_n$.

Cyman–Dzido–Lapinskas–Lo conjecture. For every fixed $k \geq 5$,

$$\lim_{n \rightarrow \infty} \frac{\tilde{r}(P_k,P_n)}{n}=\frac{3}{2}.$$

The conjecture asserts that the lower-bound strategy of Cyman, Dzido, Lapinskas and Lo is asymptotically optimal. It concerns the leading-order growth of online Ramsey numbers when one path is fixed and the other grows; the supplied source does not state whether the conjecture has been resolved.