The universal cork conjecture

Work in the category of smooth, compact, oriented $4$-manifolds with boundary. A cork is a pair $(C,f)$ where $C$ is a contractible $4$-manifold and $f:\partial C\to\partial C$ is an involution that does not extend to a diffeomorphism of $C$. An exotic pair of simply-connected, closed $4$-manifolds is a pair $(X_0,X_1)$ with $X_0\approx X_1$ and $X_0\not\cong X_1$. The pair is related by $(C,f)$ if there is an embedding $\mathbb{e}:C\to X_0$ such that

$$X_0\setminus \operatorname{int}(\mathbb{e}(C))\bigcup_{\mathbb{e}\circ f}C\cong X_1.$$

A cork is universal if it relates every exotic pair of simply-connected, closed $4$-manifolds.

Universal cork conjecture. There is no universal cork, i.e. no cork $(U,f_U)$ relating every exotic pair of simply-connected, closed $4$-manifolds.

Every exotic pair of simply-connected, closed $4$-manifolds is related by some cork, but it is not known whether a single cork can relate all such pairs. The conjecture asserts that no such universal cork exists; in particular, it would imply that the Akbulut cork is not universal.