The generalized Andrews–Curtis conjecture for 2-dimensional CW-pairs

Let $(X,Y)$ and $(X',Y)$ be 2-dimensional CW-pairs. They are simple-homotopy equivalent if there is a map $X\to X'$ homotopic to a deformation and inducing the identity on the common subcomplex $Y$; they are 2-equivalent if one can be deformed to the other by homotopies of attaching maps and expansions and collapses of dimension at most $2$, restricting to the identity on $Y$.

Generalized Andrews–Curtis conjecture. Any two simple-homotopy equivalent 2-dimensional CW-pairs are 2-equivalent.

For dimensions other than $2$, results of Wall show that simple-homotopy equivalence implies $n$-equivalence for $n$-dimensional CW-pairs. The dimension-two case is the generalized Andrews–Curtis conjecture, and it is believed to be false.