The kernel conjecture for isotopy extension and barbell implantations

Let $\operatorname{Emb}(S^1,S^1\times S^3;S^1_0)$ be the based embedding space, and let $\operatorname{Diff}(S^1\times B^3\operatorname{fix}\partial)$ and $\operatorname{Diff}(B^4\operatorname{fix}\partial)$ denote the corresponding diffeomorphism groups. Isotopy extension induces a map

$$p:\pi_1 \operatorname{Emb}(S^1,S^1\times S^3;S^1_0)\longrightarrow \pi_0\left(\operatorname{Diff}(S^1\times B^3\operatorname{fix}\partial)/\operatorname{Diff}(B^4\operatorname{fix}\partial)\right).$$

For the relevant invariant $W_2$, kernel conjecture. The kernel of $p$ is the subgroup with $W_2=0$. In particular, the implantations $\beta_{\theta_k}$ for $k\geq 2$ and $\beta_{\alpha_k}$ for $k\geq 1$ are isotopically nontrivial. This would identify the precise obstruction detected by isotopy extension and establish infinitely many nontrivial implantation classes.