The $\E_\infty$ Thom spectrum conjecture for $\mathrm{tmf}_1(3)$

Let $Mj$ be the $\mathcal{E}_\infty$ Thom spectrum obtained from the infinite loop map

$$\mathrm{Q}(S^2\vee B\mathrm{Spin}^{[6]}\vee B\mathrm{O}\langle8\rangle^{[15]})\to B\mathrm{Spin}^{\mathrm{c}}\to B\mathrm{SO}.$$

Let $Mj/\!/w_4,w_8,w_{12}$ be the $\mathcal{E}_\infty$ $3$-cell complex obtained from $Mj$ by attaching $\mathcal{E}_\infty$ cells of dimensions $5$, $9$ and $13$ along maps $w_4,w_8,w_{12}$.

$\mathcal{E}_\infty$ Thom spectrum conjecture. There is an $\mathcal{E}_\infty$ morphism $Mj\to\mathrm{tmf}_1(3)$ which factors through $Mj/\!/w_4,w_8,w_{12}$, and the induced morphism

$$Mj/\!/w_4,w_8,w_{12}\to\mathrm{tmf}_1(3)$$

induces an epimorphism on $H_*(-)$ which is an isomorphism up to degree $15$.

This proposes an $\mathrm{E}_\infty$ Thom-spectrum model related to $\mathrm{tmf}_1(3)$, extending the constructions associated with $Mj^{\mathrm{c}}$ and $k\mathrm{U}$. The existence of the factorization and its stated homology properties remain open.