The boundary Dehn twist conjecture for the Case 2 Cayley-fibration fiber

Let $F$ be a smooth manifold homeomorphic to

$$K3\# K3 \# (S^2 \times S^2).$$

Let $B\subset F$ be a small $4$-ball. A boundary Dehn twist is the boundary-supported diffeomorphism of $F\setminus B$ associated with a Dehn twist of its boundary; it is nontrivial when it represents a nontrivial mapping class. Boundary Dehn twist conjecture. The manifold $F\setminus B$ admits a nontrivial boundary Dehn twist. This conjecture concerns the fiber topology arising in Case 2 of the paper's classification theorem and is closely related to the open question whether the corresponding manifolds satisfy Property (S); its status is unresolved.