Exponential gadget conjecture for induced odd cycles

Let $k$ be a positive integer. Exponential gadget conjecture. There is a graph $G$ with $e^{O(k)}$ edges such that every $k$-coloring of its edges contains a monochromatic odd cycle of length at least $5$ as an induced subgraph. Such a gadget would suffice, via the paper's construction, to obtain the conjectured exponential bound for induced size-Ramsey numbers of odd cycles. The existence of this gadget graph is left open.