Chromatic redshift conjecture for complex and quaternionic Mahowald invariants

Let $Y$ be a type $n$ finite complex with a $v_n$-periodic self-map $v$, and let $d_{\mathbb{K}}$ denote the relevant dimension for $\mathbb{K}$. Suppose that $\alpha \in \pi_*(\Sigma^{-d_{\mathbb{K}}}Y)$ is $v$-periodic. Let $M_{\mathbb{K}}(\alpha)$ be the corresponding complex or quaternionic Mahowald-invariant coset.

Chromatic redshift conjecture. If $\alpha \notin M_{\mathbb{K}}(\alpha)$, then $M_{\mathbb{K}}(\alpha)$ consists entirely of $v_n$-torsion elements. If $w:\Sigma^dY\to Y$ is a power of $v$ annihilating every element of $M_{\mathbb{K}}(\alpha)$, and $Z=\operatorname{cofib}(w)$, so that $Z$ has type $n+1$ and every element of $M_{\mathbb{K}}(\alpha)$ extends to a map from $Z$ to a suitable sphere, then at least one of these maps is $v_{n+1}$-periodic.

This is proposed as a generalization of the Mahowald–Ravenel redshift conjecture. The surrounding discussion says that the earlier conjecture is supported by low-height computations, while the proposed complex and quaternionic generalization remains open.