In the paper "An-type singularity and nondisplaceable Lagrangian tori", https://arxiv.org/pdf/1710.11221.pdf, by Sun the author proves that when taking the toric degeneration of a semi-Fano toric surface the potential function of an interior fiber of the smoothing of the degeneration is the limit of the potential functions, i.e., suppose that we have semi-Fano toric surfaces $\Delta_{\alpha}$ and its toric degeneration $\Delta=\lim_{\alpha \rightarrow 0}\Delta_{\alpha}$ a toric orbifold with $A_n$-type singularities which is obtained by collapsing the $-2$-spheres of $\Delta_{\alpha}$. Then the authors prove that if we consider the potential functions $\mathfrak{P}\mathfrak{D}_{\alpha}$ then the potential function of an interior point of the smoothing of the $A_n$-singularities of $\Delta$ is $\lim_{\alpha \rightarrow 0}\mathfrak{P}\mathfrak{D}_{\alpha}$. Looking at the authors proof the fact that we are collapsing all of the $-2$-spheres on $\Delta_{\alpha}$ is very important to not have bubbling of maslov disks of index $0$. My question is if there is some sort of generalization of this results were we collapse some of the $-2$-spheres but not all of them, is there a way to compute the potential function of an interior point in these cases?
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