The 0-, 1-, 2-handle conjectures for contractible manifolds

Let $\Delta^4$ be a Poincare ball, and let $\Delta^5$ be a contractible 5-manifold. A handle decomposition has only 0-, 1- and 2-handles when all its handles have these indices. 0-, 1-, 2-handle conjectures. The following two assertions hold: (i) If $\Delta^4$ is a Poincare ball, then $\Delta^4\times I$ has a handle decomposition with only 0-, 1- and 2-handles. (ii) If $\Delta^5$ is contractible and built from 0-, 1- and 2-handles, then $\Delta^5=B^5$. These are proposed as a classical approach to the Poincare ball 5-ball conjecture and are open problems in high-dimensional smooth topology.