The no-BPS finite stabilization conjecture

Let $(X,\phi)$ be spectral data with no BPS states, and let $Z_i$ be the sequence of two-manifold constructions produced by the modification process. A fold marking is a marking in the scaffolding indicating a fold line.

No-BPS finite stabilization conjecture. The sequence $Z_i$ is well-defined until it reaches a construction with no more fourfold points or sixfold points with triple fold lines; the process stops in finite time, at least locally on $X$; and at the end there are no more fold markings in the scaffolding.

This is the central finiteness and termination claim underlying construction of the nonpositively curved core and the resulting pre-building. The source gives no resolution of it.