The flow category moves conjecture

A framed flow category is a flow category equipped with framing data, and its associated stable homotopy type is the stable homotopy invariant constructed from that framing. Two framed flow categories are move equivalent if they are related by a finite sequence of flow category moves: perturbation, stabilization, handle cancellation, and the extended Whitney trick.

Flow category moves conjecture. If two framed flow categories determine the same stable homotopy type, then they are move equivalent.

These moves preserve the associated stable homotopy type and model operations familiar from Morse theory. The conjecture asserts that they provide a complete calculus for framed flow categories, but the supplied source does not state a resolution.