The square-root query-complexity conjecture for testing H-minor freeness

Let $H$ be a fixed graph. In the bounded-degree graph model, a graph is $H$-minor free if it contains no minor isomorphic to $H$.

Square-root query-complexity conjecture. For every $H$, being $H$-minor free can be tested with one-sided error using $\tilde{O}(\sqrt{n})$ queries, where $n$ is the number of vertices.

The paper notes an $\Omega(\sqrt{n})$ lower bound for testing $H$-minor freeness with one-sided error when $H$ is not a tree, so the conjecture asserts that this lower bound is tight up to polylogarithmic factors. The broader status of the conjecture is not specified in the source.