The bounded-region finiteness conjecture

Let $(X,\phi)$ be spectral data with no BPS states, and let the associated sequence of two-manifold constructions be restricted to a bounded region of $X$.

Bounded-region finiteness conjecture. On bounded regions, the process stabilizes in finitely many steps.

This is the finiteness assertion associated with the main no-BPS stabilization conjecture and is used to obtain a well-defined pre-building on bounded regions. The source gives no resolution of the claim.