We know $L-L_{0}$ is homologous to zero in $X$, and contained in a bounded subset of $X$ by Cor. 5.13. By a version of the isoperimetric theorem (cf. Prop. 5.11), we can find some compactly supported integral current $\mathcal{C}$ with $\partial\mathcal{C}=L-L_{0}$, with mass bound

$$\text{Mass}(\mathcal{C})\leq\text{const}\cdot\min\{\text{Mass}(L-L_{0})^{(n+1)/n},\text{Mass}(L-L_{0})\}.$$

This $\mathcal{C}$ has no relation to holomorphic curves. The quantitative almost calibratedness implies a volume bound on $L$ (cf. Lemma 2.1), hence $\text{Mass}(\mathcal{C})$ is a priori bounded. The homological nature of the Solomon functional (20) gives

$$\mathcal{S}(L)=\int_{L}f_{L}\text{Im}(e^{-i\hat{\theta}}\Omega)-\int_{L_{0}}f_{L_{0}}\text{Im}(e^{-i\hat{\theta}}\Omega)-\text{Im}\int_{\mathcal{C}}\lambda\wedge e^{-i\hat{\theta}}\Omega.$$

The mass bound then implies

$$|\int_{\mathcal{C}}\lambda\wedge e^{-i\hat{\theta}}\Omega|\leq C\left\lVert\lambda\right\rVert_{C^{0}}\left\lVert\Omega\right\rVert_{C^{0}}\text{Mass}(\mathcal{C})\leq\text{const}.$$

Here since $L$ is contained in a bounded region, the terms $\lambda$ and $\Omega$ are bounded. Finally, using the potential clustering bound,

$$\begin{split}&|\int_{L}f_{L}\text{Im}(e^{-i\hat{\theta}}\Omega)-\int_{L_{0}}f_{L_{0}}\text{Im}(e^{-i\hat{\theta}}\Omega)-\bar{\mathcal{S}}(L)|\\ &\leq A(\int_{L}|\text{Im}(e^{-i\hat{\theta}}\Omega)|+\int_{L_{0}}|\text{Im}(e^{-i\hat{\theta}}\Omega)|)\\ &\leq A(\text{Mass}(L)+\text{Mass}(L_{0}))\leq\frac{2A}{\sin\epsilon}\int_{L_{0}}\text{Re}\Omega.\end{split}$$

Combining the above shows the a priori bound on $|\mathcal{S}(L)-\bar{\mathcal{S}}(L)|$. ∎