The epimorphism conjecture for the $\E_\infty$ orientation from $Mj_3$ to $\mathrm{tmf}$

Let $Mj_3$ be the indicated $\mathcal{E}_\infty$ Thom spectrum, let $\mathrm{tmf}$ be connective topological modular forms, and suppose that $Mj_3\to\mathrm{tmf}$ is its $\mathcal{E}_\infty$ orientation. A ring epimorphism is a surjective homomorphism of homotopy rings.

Epimorphism conjecture. The $\mathcal{E}_\infty$ orientation $Mj_3\to\mathrm{tmf}$ induces a ring epimorphism

$$\pi_*(Mj_3)\to\pi_*(\mathrm{tmf}).$$

Analogous epimorphisms are known for the cases $Mj_1$, $Mj_2$ and $Mj^{\mathrm{c}}$. The assertion for $Mj_3$ is related to, and implied by, the proposed module-spectrum splitting, but remains open here.