The ordered-forest core conjecture for polynomial testability

Let $F$ be an ordered graph, meaning a graph equipped with a linear order on its vertices. An ordered forest is an ordered graph whose underlying graph is a forest, and the core of $F$ is its smallest retract under order-preserving graph homomorphisms. Let $q_{F\text{-free}}(\varepsilon)$ denote the query complexity of testing $F$-freeness. Ordered-forest core conjecture.

$$q_{F\text{-free}}(\varepsilon)=\operatorname{poly}(1/\varepsilon)\quad\text{if and only if the core of }F\text{ is an ordered forest}.$$

The conjecture was posed by the first author and Tomon. It seeks a characterization of polynomial testability for ordered graph properties; the source states that it remains open.