The inertia-group conjecture for 4-dimensional complete intersections

Let $X_4(\underline d)$ be a $4$-dimensional complete intersection, let $v_2(X_4(\underline d))$ and $v_4(X_4(\underline d))$ denote its Wu classes, regarded as elements of $\mathbb{Z}/2$, and let $p_1(4,\underline d)$ be the integer determined by its first Pontryagin class. Write $I(X_4(\underline d))\subseteq\Theta_8\cong\mathbb{Z}/2$ for the inertia group, consisting of homotopy $8$-spheres whose connected sum with $X_4(\underline d)$ is diffeomorphic to $X_4(\underline d)$. The inertia-group conjecture. The inertia groups are given by

$$\begin{array}{c|c|c|c} v_2(X_4(\underline d)) & v_4(X_4(\underline d)) & p_1(4,\underline d)\!\!\mod~8 & I(X_4(\underline d)) \\ \hline\hline 0 & - & - & 0 \\ \hline 1 & 0 & - & \Theta_8 \\ \hline 1 & 1 & 3 & 0 \\ \hline 1 & 1 & 7 & \Theta_8 \end{array}$$

Here a dash means that the corresponding invariant is irrelevant in that case. The first two cases are established earlier in the paper, while the third and fourth cases concern the previously unknown non-spin situation and would determine whether the exotic $8$-sphere acts trivially on the diffeomorphism type.