Strong sparse non-liftability conjecture for chain complexes

A chain complex over $\mathbb{F}_2$ is sparse when every row and column of every boundary operator has $O(1)$ nonzero entries; a sparse lift to $\mathbb{Z}$ has uniformly bounded row and column $1$ norms. A $2$-complex has cells in degrees $0$, $1$, and $2$. Strong sparse non-liftability conjecture. There exist sparse chain complexes over $\mathbb{F}_2$ with no sparse lift to integer coefficients, and there exist sparse $2$-complexes over $\mathbb{F}_2$ with no sparse lift to integer coefficients. This conjecture asserts that the obstruction already occurs for general sparse complexes and, more strongly, for sparse $2$-complexes; the paper presents it as open.