The kernel-generator conjecture for self-dual VOSAs

Let $A_{-2c}$ be the kernel of

$$\Phi\colon \pi_{-2c}\operatorname{TMF}\to \pi_{-2c}\operatorname{KO}((q)).$$

For $c=32,31/2,15,14$, the kernel is respectively isomorphic to $\mathbb{Z}/3,\mathbb{Z}/2,\mathbb{Z}/2,\mathbb{Z}/2$. The kernel-generator conjecture. The generator of $A_{-2c}$ comes from the corresponding self-dual VOSA $\mathbb{V}$ in the source's table. In particular, it lifts to

$$\operatorname{TMF}^{2c+t}_G(\mathrm{pt})$$

for suitable $G$ and $t\colon BG\to K(\mathbb{Z},4)$, with nonzero image in $\operatorname{KO}^{2c}_G((q))(\mathrm{pt})$ given by $\eta(q)^{-2c}[\mathbb{W}]$, where $\mathbb{W}$ is the canonically $\mathbb{Z}/2$-twisted module of $\mathbb{V}$. This conjecture proposes a uniform explanation for the observed kernel classes, extending the specific examples discussed earlier in the paper.