Unavoidable torsion in sparse lifts conjecture

A sparse chain complex over $\mathbb{F}_2$ has a sparse lift to integer coefficients when its boundary operators can be lifted while remaining sparse. Torsion obstruction conjecture. There exist sparse chain complexes over $\mathbb{F}_2$ that have a sparse lift to integer coefficients, but every sparse integer lift has some torsion in its homology and cohomology. Moreover, there exist sparse $1$-complexes with trivial homology and cohomology for which every sparse integer lift has some torsion in homology and cohomology. This would exhibit torsion as an unavoidable obstruction even when a sparse lift exists and the original complex is homologically trivial; the conjecture remains open.