Uniform polynomial Chowla conjecture over finite fields

Let $q$ be a fixed power of an odd prime and let $c>0$ be fixed. For any polynomial $F\in\mathbb{F}_q[x][T]$ that is separable in the variable $T$, define the Liouville function by $\lambda(g)=(-1)^k$ when $g=P_1\cdots P_k$ is the prime factorization of $g\in\mathbb{F}_q[x]$. Uniform polynomial Chowla conjecture.

$$\sum\limits_{h\in\mathbb{F}_q[y]\,\mathrm{monic}\atop{\deg h=n}}\lambda(F(y,h(y)))=o(q^n),\quad n\to\infty$$

uniformly in $F$ satisfying $\deg_y(F)\leq cn$ and $\deg_T(F)\leq c$. This conjecture is a uniform finite-field analogue of Chowla's conjecture on cancellation in the Liouville function; in the paper it is used as a conditional input for distinguishing $S_n$ from $A_n$ in the small-box model.