A short exposition of the Levine-Lidman example of spineless 4-manifolds

A 2018 paper by A. Levine and T. Lidman outlines a proof of the following interesting result in topology of manifolds: there is a compact smooth 4-manifold $W$ with boundary such that $W$ is homotopy equivalent to $S^2$ but there does not exist an embedding $S^2\to W$ which is a homotopy equivalence and is simplicial for some triangulations of $W$ and of $S^2$. We present a shorter (and hopefully clearer) exposition. We reveal that some parts of the proof are missing, and that some results are used in that paper without proof or reference, or even without explicit statement.