On the boundary algebras of the Jacobian algebras of the bordered marked surfaces

Given $\sigma$ a triangulation of bordered surface with marked points and punctures $(S, M)$, we associate an ice quiver with potential $(Q_{\sigma}, W_{\sigma}, F)$ and define the corresponding Jacobian algebra $\Gamma_{\sigma}$. We show that the boundary algebra $B(\sigma)$ of $\Gamma_{\sigma}$ depends only on the surface $(S, M)$.