The pochette surgery conjecture for homotopy 4-spheres

Let $e$ be an embedding of a pochette into $S^4$, let $S_e$ denote its associated spun 2-knot, and let $S^4(e,p/q,\varepsilon)$ denote the result of pochette surgery with parameters $p/q$ and $\varepsilon$. The pochette surgery conjecture. If $S^4(e,p/q,\varepsilon)$ is homotopy equivalent to the 4-sphere $S^4$, then $S^4(e,p/q,0)$ is diffeomorphic to $S^4$ and $S^4(e,p/q,1)$ is the Gluck surgery along $S_e$. In particular, if $S_e$ is a twist-spun 2-knot or a 0-slice 2-knot, then $S^4(e,p/q,\varepsilon)$ is diffeomorphic to $S^4$ for every $\varepsilon\in\{0,1\}$. The conjecture relates homotopy 4-spheres obtained by pochette surgery to the smooth 4-dimensional Poincaré problem and to Gluck surgery; the source provides no resolution of the general claim.