Andrews–Curtis conjecture for finite contractible 2-complexes

Let $L$ be a finite, contractible 2-complex. A 3-deformation is the operation, described in the source, of successively gluing 3-balls along open 2-cells and performing inverse operations. Andrews–Curtis conjecture. $L$ 3-deforms to a single vertex. The conjecture is a presentation-theoretic and topological strengthening relevant to finite contractible 2-complexes; the source states it as the assertion that every such complex can be reduced to a point by 3-deformations, without providing a resolution.