Given small numbers $0<\lambda,\kappa\ll 1$, then for $t$ small enough depending on $\lambda,\kappa$, there exist appropriately chosen local potentials $\phi_{CY,J,t}$ on $\text{Log}_{\mathcal{X}}^{-1}(\text{Int}(\Delta_{J}))$, normalized to $\inf(\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}})=0$, satisfying

$$d\mu_{t}(\{\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}\geq\kappa/4\})<\lambda,$$

and $\left\lVert\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}\right\rVert_{L^{\infty}}\leq C$ independent of $\lambda,\kappa$ and small $t$.