We have already made extensive comparisons between the variational approach and Joyce’s LMCF program, but it may help to summarize a few highlights.

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  Joyce’s program is much more ambitious in that it tackles the entire derived Fukaya category, not just the almost calibrated Lagrangians. We feel the quantitative almost calibratedness is so pervasively used in the variational approach that it cannot be removed. Dropping the almost calibratedness will give rise to significantly more difficulties in Joyce’s program: the collapsing of zero objects can then happen, and the Solomon functional no longer needs to decrease. Neves’s example of finite time singularity [62] is a concrete manifestation of the difficulty. The almost calibrated condition is also natural from the viewpoint of the continuity method (cf. section 4.2), which deals with special Lagrangians inside varying ambient almost Calabi-Yau structures.

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  Joyce does not specify the Bridgeland stability in a priori Floer theoretic terms. An a priori guess on the nature of the stability condition is central to the variational method. Even though our picture is largely conjectural, it seems to be the most precise description hitherto of how stability condition comes into the existence questions of special Lagrangians.

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  Joyce primarily focuses on compact Calabi-Yaus, and mentions the exact case only as an easier analogue. We have focused on the exact case, although we feel some parts of our picture may extend to compact Calabi-Yaus, if one is prepared to overcome (even more) significant Floer theoretic technical hurdles. However, we do not know what would replace the a priori estimates on the Lagrangian potentials, and notably the potential clustering condition.

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  Joyce’s LMCF involves objects with a priori higher regularity, even though its infinite time convergence behaviour may well require understanding weak regularity Lagrangians. The variational method requires working with varifold/current like objects throughout.

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  Joyce’s LMCF needs to make essential use of genericity conditions. This in particular requires extremely precise classification of all possible generic singularities in order to perform surgeries, a task that becomes overwhelmingly difficult for complex dimension $\geq 3$. Our variational program is less sensitive to such arguments. On the other hand, we still potentially need to understand some generic singularities, so that the class $\mathcal{L}$ contains enough competitors, to enable the proof of the $L^{p}$-smoothing property for some $p\geq 1$, and Conjecture 5.23.

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  Although time and again we appealed to Joyce’s LMCF to heuristically justify certain claims, it is only because we lack other ways of constructing Lagrangian competitors with sufficient control, and the basic logical framework of the variational approach is independent of the LMCF. It seems desirable (on account of the extraordinary difficulty of Joyce’s program) to keep this logical independence manifest in the program to rigorize our variational proposal.

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  Joyce’s program has a number of highly nontrivial categorical predictions discussed in section 3.6, such as the idempotent closedness of $D^{b}Fuk(X)$. Even if these predictions turn out to be false, it would not affect the validity of the variational method.