A numerical invariant for homotopy Hopf manifolds

Let $\sigma^8$ denote the space of homotopy Hopf manifolds, and let $\sigma^8_{\textrm{Milnor}}$ denote its Milnor subspace. A numerical map is a map

$$N:\sigma^8\longrightarrow\mathbb R.$$

Numerical-map conjecture. There is a numerical map $N$ which is constant with value $\frac{1}{12}$ on $\sigma^8_{\textrm{Milnor}}$.

The conjecture is motivated by the surjectivity of the homotopy-group map induced by a multiplicative map from $MO\langle 8\rangle$ to $\mathrm{tmf}$ with underlying genus the Witten genus, together with the asserted identification of $\pi_*\mathrm{tmf}$ and $\pi_*S^0$.