Exponential induced size-Ramsey conjecture for odd cycles

Let $C_n$ be the cycle on $n$ vertices, and let $\hat{r}^k_{\operatorname{ind}}(C_n)$ denote the smallest number of edges in a graph whose every $k$-coloring contains a monochromatic copy of $C_n$ as an induced subgraph. Exponential induced-cycle conjecture. For odd $n$,

$$\hat{r}^k_{\operatorname{ind}}(C_n)\leq e^{O(k)}n.$$

The paper proves the weaker bound $\hat{r}^k_{\operatorname{ind}}(C_n)\leq e^{O(k\log k)}n$ for odd cycles, so the conjectured exponential dependence on $k$ remains open.