Stable sparse non-liftability conjecture for 2-complexes

A sparse $2$-complex over $\mathbb{F}_2$ is a chain complex in degrees $0$, $1$, and $2$ whose boundary operators are sparse. A stable lift is obtained by adding finitely many contractible cell pairs $b_j,b_{j+1}$, with $\partial b_{j+1}=b_j$, $\partial b_j=0$, and $\partial^T b_{j+1}=0$, and then lifting the stabilized complex to integer coefficients. Stable sparse non-liftability conjecture. There exist sparse $2$-complexes over $\mathbb{F}_2$ that have no sparse stable lift to integer coefficients. This is stronger than ordinary non-liftability because stabilization by finitely many contractible pairs is also allowed; the paper leaves the claim open.