(Heuristic) First, we claim that for a minimizing sequence $L^{(k)}$ of the Solomon functional, without loss of generality the Lagrangian potential $f_{L}^{(k)}$ is a priori bounded:

Consider the potential clustering setup. We can adjust the Lagrangian potentials on $L_{i}$ by constants separately, and as long as $\sup_{L_{j}}f_{L_{j}}\leq\inf_{L_{i}}f_{L_{i}}$ for $j<i$, this process will not affect the Novikov positivity requirement, so the Lagrangian branes should remain in $\mathcal{L}$. We view $\sup_{L_{1}}f_{L_{1}},\sup_{L_{2}}f_{L_{2}}-\sup_{L_{1}}f_{L_{1}},\ldots,\sup_{L_{N}}f_{L_{N}}-\sup_{L_{N-1}}f_{L_{N-1}}$ as independent constants. Adjusting all potentials by a common constant does not affect the Solomon functional, but allows us to set $\sup_{L_{1}}f_{L_{1}}=0$. Decreasing $\sup_{L_{i}}f_{L_{i}}-\sup_{L_{i-1}}f_{L_{i-1}}$ subject to the Novikov positivity requirement will decrease the elementary functional (58), crucially because of the semistability condition (61). The part $\mathcal{S}-\bar{\mathcal{S}}$ is unchanged. Thus after this adjustment, the sequence is still minimizing for the Solomon functional. We can thus achieve $\sup_{L_{i-1}}f_{L_{i-1}}=\inf_{L_{i}}f_{L_{i}}$ for all $i$. By the potential clustering property, we then have (62).

Next we need the compactness from geometric measure theory. As discussed in section 5.1 and 5.2, under quantitative almost calibratedness there is an a priori volume bound, and the Lagrangians all remain in a fixed bounded subset of $X$, so Federer-Fleming compactness (cf. Theorem 5.2) holds automatically. The uniform potential bound (62) would then justify that the weak limit is an almost calibrated Lagrangian current $L$ with bounded potential $f_{L}$ (cf. Lemma 5.7). The continuity of the Solomon functional (cf. Lemma 5.8) then shows $\mathcal{S}(L)=\inf_{\mathcal{L}}\mathcal{S}$.

In section 5.3 we presented the evidence for the conjectural $L^{2}$-smoothing property, which would allow us to assume a uniform a priori bound on the minimizing sequence

so we can use Allard compactness theorem 5.3. In effect, we can assume the minimizing sequence converges subsequentially both as currents and as varifolds. By assumption the class $\mathcal{L}$ is closed under the varifold/current topology of the Lagrangian, so the limit $L$ lies in $\mathcal{L}$, whence provides a minimizer in $\mathcal{L}$. ∎