A factorization conjecture for the numerical map on homotopy Hopf manifolds

Let $\sigma^8$ be the space of homotopy Hopf manifolds, let $\mathcal L$ and $\mathrm{p}_1$ be the maps appearing in the source, and let $MO\langle 8\rangle$ be the Thom spectrum of $BO\langle 8\rangle$. Write $\pi_3^S$ for the third stable homotopy group of spheres. Factorization conjecture. The map

$$\mathcal L\circ\mathrm{p}_1:\sigma^8\longrightarrow\pi_3^S$$

induces a map

$$\mathfrak{L}_3:\pi_3MO\langle 8\rangle\longrightarrow\pi_3^S$$

that can be seen as a factor of the ring isomorphism $\pi_*\mathrm{tmf}\cong\pi_*S^0$. Restricting the domain of $\mathfrak{L}_3$ properly, it can be identified as a constant function with the term $\frac{1}{12}$ in the Fourier expansion of the $\wp$-Weierstrass function.

The conjecture refines the proposed numerical invariant by relating it to stable homotopy and the Witten-genus-based map through $\mathrm{tmf}$. The supplied text does not state whether this conjecture has been resolved.