The external stabilization conjecture for exotic 4-manifolds

Let $(X_{0},X_{1})$ be an exotic pair of closed, simply-connected smooth $4$-manifolds, meaning that they are homeomorphic but not diffeomorphic. Let $X\#Y$ denote the connected sum of smooth $4$-manifolds. External stabilization conjecture. Then

$$X_{0}\#(S^{2}\times S^{2})\text{ is diffeomorphic to }X_{1}\#(S^{2}\times S^{2}).$$

This asks whether one external stabilization by $S^{2}\times S^{2}$ eliminates the smooth distinction between every exotic pair of closed simply-connected $4$-manifolds. The supplied text identifies this as a conjecture but gives no resolution evidence, so its status is recorded as open.