Short Sums of the Liouville Function over Function Fields

Let $\lambda$ denote the Liouville function for function fields. We prove that for a fixed $q$, given $h\ll \sqrt{N}$ and $h(N)\to \mathrm{\infty }$ arbitrarily slowly as $N\to \mathrm{\infty }$, then \begin{equation*} \frac{1}{q^N}\sum_{G_0 \in $\mathcal{M}$N}|\sum{G \in $\mathcal{I}$_{h}(G_0)}λ(G)|^2 \ll_q \frac{N^5}{h^2}q^{h}. \end{equation*} The proof follows a similar method of an analogous case in the integer setting developed by Chinis, adapting methods originally developed by Matomäki and Radziwiłł.