It remains to show

Prop. 6.4 says that the local potential of $\omega_{\psi,t}$ and $\phi_{0}\circ\text{Log}_{\mathcal{X}}$ differ negligibly in the $t\to 0$ limit in the $C^{0}$-sense. Ideally, one would like to use some version of $L^{1}$-volume stability to conclude the $C^{0}$-smallness of the relative potential between $\omega_{\psi,t}$ and $\omega_{CY,t}$. Unfortunately, due to the difficulty of regularization in the non-generic region, there is very little control on the volume density of $\omega_{\psi,t}$ in the non-generic region, and we cannot conclude a Skoda type estimate like (11) for $\omega_{\psi,t}$. This technical problem causes an asymmetry between $\omega_{\psi,t}$ and $\omega_{CY,t}$, and only ‘one half’ of the $L^{1}$-volume stability estimate (cf. section 4.3) applies, which is why we designed Theorem 4.7.

After the dust settles, Theorem 4.7 implies that $\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}$ concentrates near its minimum value (normalized to be zero) on a subset with almost $100\%$ of the Calabi-Yau measure (cf. [53, Prop 4.4, Cor. 4.6]). More precisely, for any given small number $\kappa,\lambda\ll 1$, then for sufficiently small $t$, the measure

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

(21)

On a slightly shrinked version of $U_{J,t}$, this can be improved to the $C^{0}$-control

$$0\leq\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}<\kappa,$$

by a slightly tricky application of the mean value inequality (cf. [53, Thm. 4.7]). Since $\kappa$ is arbitrary, this achieves Prop. 6.5, which verifies the hypothesis of Prop. 6.5, whence the weak metric version of the SYZ conjecture.