The Gottlieb group conjecture for spheres in dimension difference eight

Let $\mathbb S^n$ be the $n$-sphere, let $\iota_n\colon \mathbb S^n\to\mathbb S^n$ denote its identity map, and let $\eta_n\sigma_{n+1}\in\pi_{n+8}(\mathbb S^n)$ be the homotopy class appearing in the statement. The Gottlieb group $G_{n+8}(\mathbb S^n)$ consists of those elements of $\pi_{n+8}(\mathbb S^n)$ whose Whitehead product with $\iota_n$ vanishes.

Proposed conjecture. For $n\equiv 6\pmod 8$ with $n\geq 14$,

$$[\iota_n,\eta_n\sigma_{n+1}]=0$$

and

$$G_{n+8}(\mathbb S^n)=\{\eta_n\sigma_{n+1}\}\cong\mathbb Z_2.$$

This would determine the relevant Gottlieb group in the remaining congruence class not settled by the preceding proposition. The supplied text does not state a resolution of the proposed claim.