The induced $C_4$-freeness polynomial testability conjecture

Let $C_4$ denote the cycle with four vertices, and let $q_{\text{ind-}C_4\text{-free}}(\varepsilon)$ be the query complexity for testing induced $C_4$-freeness. Induced $C_4$-freeness conjecture.

$$q_{\text{ind-} C_4\text{-free}}(\varepsilon)=\operatorname{poly}(1/\varepsilon).$$

This is the remaining case in the characterization of graphs with polynomially testable induced-freeness, following the known results for the other graphs.