Equivariant Mahowald-invariant conjecture for classical compact Lie groups

Let $G$ be one of the compact Lie groups $SO(n)$, $SU(n)$, or $Sp(n)$. Let $M^G:\pi_*^s\rightsquigarrow\pi_*^s$ be the $G$-equivariant Mahowald invariant constructed using the corresponding non-nilpotent Euler class, and let $\alpha$ and $\beta$ be stable homotopy classes represented by framed homotopy spheres.

Equivariant Mahowald-invariant conjecture. If

$$\beta\in M^G(\alpha),\qquad \beta\neq\alpha,$$

then the homotopy sphere corresponding to $\beta$ admits a smooth $G$-action whose fixed points are the homotopy sphere corresponding to $\alpha$.

This would extend the relationship between Mahowald invariants and smooth actions already established in the paper for the real, complex, and quaternionic cases. The claim is proposed for the groups listed above and remains open in the source.