Jacobi algebra presentations for fundamental group algebras

We prove a special case of a conjecture of Davison which pertains to superpotential descriptions of fundamental group algebras $k[\pi_1(X)]$. We consider the case in which the manifold $X$ is the mapping torus $M_{g, \varphi}$ of a genus $g$ Riemann surface $\Sigma_g$ and a finite order automorphism $\varphi$, and the superpotential structure is given by the Jacobi algebra of a quiver with potential.