This question is a reformulation of a special case of the question An $\ell$-adic local system which is trivial on every fiber of a morphism (this special case did not receive an answer on MathOverflow).

Let $\mathbb A^2$ be the affine plane over the algebraic closure of the finite field $\bar {\mathbb F}_q$. Let $L$ be an $\ell$-adic local system on this $\mathbb A^2$ which restricts to a trivial local system on every vertical line $\mathbb A^1 \subset \mathbb A^2$ (i.e. every line in the above $\mathbb A^2$ given by an equation of the form $x=c$ for $c \in \bar {\mathbb F_q}$). In other words, for any vertical line $\mathbb A^1$ as above, the restriction of the representation $\rho$ of the etale fundamental group $\pi_1^{et}(\mathbb A^2)$ (which corresponds to the local system $L$) via the group homomorphism $\pi_1^{et}(\mathbb A^1) \to \pi_1^{et}(\mathbb A^2)$ is a trivial representation of $\pi_1^{et}(\mathbb A^1)$.

Is it necessarily true that $L$ is the pullback of an $\ell$-adic local system on $\mathbb A^1$ via the projection map $\mathbb A^2 \to \mathbb A^1$ sending $(x,y)$ to $x$?