The Poincare ball 5-ball conjecture

A Poincare 4-ball is a contractible 4-manifold whose boundary is $S^3$. A Schoenflies 4-ball is a Poincare 4-ball that embeds in $S^4$. Poincare ball 5-ball conjecture. If $\Delta^4$ is a Poincare ball, then

$$\Delta^4\times I\cong B^5,$$

and hence $\Delta^4\times I$ is a Schoenflies ball. This conjecture is presented as the Poincare ball embedding approach to the smooth 4-dimensional Poincare conjecture and remains open.