Topological invariance of the grid-homology spectrum

Prove that the stable homotopy type constructed from grid diagrams whose stable homology is grid homology (the Manolescu–Sarkar spectrum) is invariant under grid moves and hence defines a knot- or link-invariant spectrum.

Background

The authors outline the construction of a spectrum associated to grid diagrams whose stable homology recovers grid homology, following Manolescu–Sarkar. For this construction to yield a topological invariant, it must be shown to be unchanged under the standard grid moves (commutation and stabilization), thereby establishing knot/link invariance.

Although the spectrum itself and many of its properties are described, the necessary invariance proof is not currently available, and the authors emphasize that resolving this would enable the extraction of additional invariants beyond homology.