Kurek–Ruciński conjecture on the online-to-size Ramsey ratio for cliques

Let $K_n$ be the complete graph on $n$ vertices. The online size Ramsey number $\tilde{r}(K_n,K_n)$ is the least number of rounds needed for Builder to force a red or blue copy of $K_n$, while the size Ramsey number $\hat{r}(K_n,K_n)$ is the least number of edges in a graph whose every red-blue edge-colouring contains a red or blue copy of $K_n$.

Kurek–Ruciński conjecture.

$$\frac{\tilde{r}(K_n,K_n)}{\hat{r}(K_n,K_n)}\to 0\text{ for } n\to\infty.$$

This conjecture asks whether the adaptive online version is asymptotically negligible compared with the classical size Ramsey number. The source describes it as still open.