Kernel and image of the surgery-induced map

Determine the kernel and image of the map \(\mathrm{ps}_k:\pi_k\mathrm{Emb}_\partial(\nu S^1,D^4\setminus\nu S^2)\to\pi_{k-1}\mathrm{Diff}_\partial(D^4)\), and determine the sizes of these two groups.

Background

The paper observes that an implanted higher barbell diffeomorphism whose extra cuff is unknotted in $D^4$ lies in the image of the map $\mathrm{ps}_k$. This map is defined from the homotopy groups of the framed embedding space of a tubular neighborhood of $S^1$ in the complement of a tubular neighborhood of $S^2$, to the homotopy groups of the boundary-relative diffeomorphism group of $D^4$, via family isotopy extension and surgery along the Hopf link $S^1\sqcup S^2$.

The paper notes that Watanabe's detection theorem implies that the rationalization of this map is surjective in the relevant setting and that its image contains many nontrivial elements. Since the source can be studied explicitly using codimension-three embedding calculus, the unresolved problem is to determine both the kernel and the image of $\mathrm{ps}_k$, including how large they are.