Unlike Thomas and Yau who based their bet primarily on the Floer theoretic or categorical aspects, which are closer to the quantum world of topological field theories, Solomon’s picture is predominantly classical, and its chief limitation comes from fixing the topological type of the Lagrangian:

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  There is no appearance of the brane structure, or the role of holomorphic curves.

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  Solomon works with exact isotopic Lagrangians, but the Thomas-Yau argument suggests it is more natural to work in a derived Fukaya category class.

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  The Solomon functional is only well defined by passing to a highly nontrivial universal cover. In the very special case where $\omega$ and $\text{Im}(e^{i\hat{\theta}}\Omega)$ are exact forms on $X$, Solomon gave a formula [75, Thm 1.3] that shows his functional is well defined on $\mathcal{O}$. We view the exactness on $\text{Im}(e^{i\hat{\theta}}\Omega)$ as too strong an assumption for applications.

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  The infinite dimensional Riemannian structure is incomplete in a much more severe way compared to the B-side analogues. This means that in non-pathological examples, we can reach the boundary of the exact isotopy class within finite distance in the Solomon metric, such that the Solomon functional remains finite.

  This is geometrically very significant. In the LMCF approach, this would strongly suggest the formation of finite time singularity, which is a major difference with the HYM case. In the variational viewpoint, this incompleteness would negate all the favourable arguments from the convexity of the functional and the non-positivity of curvature, and suggest instead that the exact isotopy class is not an adequate framework for finding special Lagrangians. We will discuss later that a more promising variational framework needs to incorporate Lagrangians from the same derived Fukaya category class, not just the same exact isotopy class.