\(\DeclareMathOperator{\LC}{LC} \DeclareMathOperator{\LS}{LS} \newcommand{\CA}{\mathcal{A}} \newcommand{\CB}{\mathcal{B}} \DeclareMathOperator{\Aut}{Aut}\)
Lemma 2.   A locally constant sheaf on a simply connected space is constant.

Proof. Let \(X\) be a simply connected space with \(F\) locally constant. Then by lemma 1 the étale space of \(F\) covers \(X\). The only covers of a simply connected space are a discretely indexed copies of itself, and hence we can write our étale space in the form \(p \colon X \times M \to X\). Since \(X\) is connected, the sections of this map are constant, meaning our sheaf is constant. \(\blacksquare\)