The a priori limit strategy does not explicitly appeal to NA geometry. Its principal remaining difficulty is the extension problem. Based on the experience with the Fermat example, we anticipate that extension to toric metrics on ambient toric varieties is a useful technique, and the problem may have a substantially combinatorial aspect. As we emphasized in section 5.6, an explicit class of convex potentials would also be essential for a direct formulation of the real MA equation, which is likely needed for more refined questions such as the affine structure and the singular set of the real MA metric on $Sk(X)$ (cf. the Kontsevich-Soibelman conjecture in section 3.3).

In contrast, the NA pluripotential theory is built around the central concept of NA semipositive metrics on $X_{K}^{an}$, which extend up to $C^{0}$-small error to Kähler potentials on $X_{t}$ via the Fubini-Study approximation. In that respect, NA pluripotential theory may be viewed as a disguised solution of the extension problem. To make contact with differential geometric applications, however, requires some additional hypothesis such as the NA MA-real MA comparison property. Comparing the difficulties in the two strategies, we speculate that proving the NA MA-real MA comparison property requires a more concrete characterization of NA semipositive metrics, perhaps of explicitly combinatorial nature.