By construction in Lemma 4.1, the local potential $\phi_{J,t}$ of $\omega_{FS,t}$ is $\epsilon$-close to $\phi_{0}\circ\text{Log}_{\mathcal{X}}$, and since $\epsilon\ll\kappa$ these two are practically the same. Up to an overall normalisation constant, which is fixed by $\inf=0$, we have $\phi_{CY,J,t}=\phi_{CY,t}+\phi_{J,t},$ so the measure bound follows from the previous result.

The uniform $L^{\infty}$ bound follows from Prop. 4.3 and Lemma 4.1 without any reference to $\lambda,\kappa$. ∎