  We study threefolds $Y$ fibred by $A_m$-surfaces over a curve $S$ of positive
genus. An ideal triangulation of $S$ defines, for each rank $m$, a quiver
$Q(\Delta_m)$, hence a $CY_3$-category $(C,W)$ for any potential $W$ on
$Q(\Delta_m)$. We show that for $\omega$ in an open subset of the K\"ahler
cone, a subcategory of a sign-twisted Fukaya category of $(Y,\omega)$ is
quasi-isomorphic to $(C,W_{[\omega]})$ for a certain generic potential
$W_{[\omega]}$. This partially establishes a conjecture of Goncharov concerning
`categorifications' of cluster varieties of framed $PGL_{m+1}$-local systems on
$S$, and gives a symplectic geometric viewpoint on results of Gaiotto, Moore
and Neitzke.
