The USPN conjecture for non-adaptive one-sided permutation-freeness testing

Let $\pi$ be a permutation of length $k$, and let $u(\pi)$ denote its USPN. A permutation is $\pi$-free if it contains no occurrence of $\pi$. A non-adaptive one-sided $\varepsilon$-test is an algorithm that always accepts $\pi$-free inputs, rejects inputs that are $\varepsilon$-far from $\pi$-free, and whose queries are fixed independently of the input.

USPN conjecture. For any permutation $\pi$ of any length, $\pi$-freeness has a non-adaptive one-sided $\varepsilon$-test making

$$\tilde{\Theta}_{\varepsilon}\left(n^{1-1/u(\pi)}\right)$$

queries.

The preceding discussion states that this would make the USPN $u(\pi)$ the correct parameter governing the difficulty of non-adaptively testing $\pi$-freeness with one-sided tests, matching the known lower bound up to factors polynomial in $\varepsilon$ and $\log n$.