Property (S) conjecture for manifolds homeomorphic to K3#K3#(S2×S2)

Let $F$ be a smooth manifold homeomorphic to

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

Such a manifold satisfies Property (S) when it has the spinnability property for fiber bundles defined earlier in the paper. Property (S) conjecture. Any smooth manifold homeomorphic to $K3\# K3 \# (S^2 \times S^2)$ satisfies Property (S). The source explicitly describes this as an open problem, closely related to exotic boundary Dehn twists of $K3\# K3 \# (S^2 \times S^2)\setminus B^4$; current technology does not establish it for every smooth manifold in this homeomorphism class.