Kirby's geometric 1-connectedness conjecture for smooth 4-manifolds

Let $M$ be a simply connected closed smooth $4$-manifold. Say that $M$ is geometrically $1$-connected if it admits a handle decomposition without $1$-handles.

Kirby's conjecture. Every simply connected closed smooth $4$-manifold is geometrically $1$-connected.

A proof would imply Živaljević's conjecture for closed manifolds via the paper's theorem relating geometric $1$-connectedness to local constructibility. The conjecture is presented as an open problem, with some partial progress known, but it is not resolved in the source.