A classification conjecture for homotopy Hopf manifolds failing the one-twelfth invariant

Let $\sigma^8$ be the space of homotopy Hopf manifolds, let $\widetilde{\sigma}$ be the map appearing in the source, and let $\theta^7$ denote the group of homotopy seven-spheres. A homotopy Hopf manifold is said to fail the one-twelfth association when it cannot be associated with the number $\frac{1}{12}$ in the Fourier expansion of the $\wp$-Weierstrass function. Classification conjecture. Given $X\in\sigma^8$, if $X$ fails the one-twelfth association, then either

1. $X$ is the image under $\widetilde{\sigma}$ of an element $(m,1-m,l)\in\pi_3(\mathrm{SO}(3))\otimes\pi_3(\mathrm{SO}(3))\otimes\pi_1(\mathrm{SO}(2))$ with $l\neq 1$; or
2. $X=\Sigma^7\times\mathrm{S}^1$ for some $\Sigma^7\in\theta^7$ that admits no realization as the total space of an $\mathrm{S}^3$-bundle over $\mathrm{S}^4$, equivalently, $\Sigma^7$ is not the total space of a Milnor bundle.

The claim is presented as a proposed classification of the exceptions to the numerical assignment by separating nontrivial twisting data from products involving homotopy seven-spheres that are not Milnor-bundle total spaces. The supplied text gives no evidence of resolution.