A Relation in the Stable Homotopy Groups of Spheres

Let $p\geqslant 7$ be an odd prime. Based on the Toda bracket $\langle\alpha_1\beta_1^{p-1}, \alpha_1\beta_1, p, \gamma_s \rangle $, the authors show that the relation $\alpha_1\beta_1^{p-1}h_{2,0}\gamma_s$$=\beta_{p/p-1}\gamma_s $ holds. As a result, they can obtain $\alpha_1\beta_1^{p}h_{2,0}\gamma_s=0 \in \pi_*(S^0) $ for $2 \leqslant s \leqslant p-2$, even though $\alpha_1h_{2,0}\gamma_s $ and $\beta_1\alpha_1h_{2,0}\gamma_s$ are not trivial. They also prove that $\beta_1^{p-1}\alpha_1h_{2,0}\gamma_3$ is nontrivial in $\pi_*(S^0) $ and conjecture that $\beta_1^{p-1}\alpha_1h_{2,0}\gamma_s$ is nontrivial in $\pi_*(S^0) $ for $3 \leqslant s \leqslant p-2$. Moreover, it is known that $\beta_{p/p-1}\gamma_3=0 \in {\rm Ext}^{5,*}_{BP_*BP}(BP_*, BP_*)$, but $\beta_{p/p-1}\gamma_3$ is nontrivial in $\pi_*(S^0)$ and represents the element $\beta_1^{p-1}\alpha_1h_{2,0} \gamma_3$.