Abstract: We study threefolds Y fibred by Am-surfaces over a curve S of positive genus. An ideal triangulation of S defines, for each rank m, a quiver Q(Δm), hence a CY3-category C(W) for any potential W on Q(Δm). We show that for ω in an open subset of the Kähler cone, a subcategory of a sign-twisted Fukaya category of (Y, ω) is quasi-isomorphic to (C, W[ω]) for a certain generic potential W[ω]. This partially establishes a conjecture of Goncharov (in: Algebra, geometry, and physics in the 21st century, Birkhäuser/Springer, Cham, 2017) concerning ‘categorifications’ of cluster varieties of framed PGLm+1-local systems on S, and gives a symplectic geometric viewpoint on results of Gaiotto et al. (Ann Henri Poincaré 15(1):61–141, 2014) on ‘theories of class S’.