It seems to me that Zeeman's collapsing conjecture satisfies the criteria given. The Zeeman conjecture implies both the Poincaré conjecture (proved in 2003) and the Andrews-Curtis conjecture.
The following is a quote from Matveev's book, where it is proved that ZC restricted to special polyhedra is equivalent to the union of PC and AC.
Theorem 1.3.58 may cast a doubt on the widespread belief that ZC is false. If a counterexample indeed exists, then either it has a “bad” local structure (is not a special polyhedron) or it is a counterexample to either AC or PC.