In the Fermat case, there is a uniform $C^{0}$-stability estimate:

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  Inside $U^{s,*}_{w}\subset X_{s}$, for $\langle m,w\rangle=1$, the local potentials satisfy $|\varphi_{CY,s,m}-\psi_{CY,s,m}|\leq Cs^{-1/2}\log s,$ or equivalently $|\varphi_{CY,s}-\psi_{CY,s}|\leq Cs^{-1/2}\log s$.

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  Inside $U^{s,face}_{w}$, the local potentials satisfy $|\varphi_{CY,s,0}-\psi_{CY,s,0}|\leq Cs^{-1/2}\log s$, or equivalently $|\varphi_{CY,s}-\psi_{CY,s}|\leq Cs^{-1/2}\log s$.