It seems to me that [Zeeman's collapsing conjecture][1] satisfies the criteria given. The Zeeman conjecture implies both the [Poincaré conjecture][2] (proved in 2003) and the [Andrews-Curtis conjecture][3].

The following is a quote from Matveev's [book][4], 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.


  [1]: https://en.wikipedia.org/wiki/Zeeman_conjecture
  [2]: https://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture
  [3]: https://en.wikipedia.org/wiki/Andrews%E2%80%93Curtis_conjecture
  [4]: https://books.google.com/books?id=vFLgAyeVSqAC&pg=PA46