Epimorphism conjecture for a characteristic morphism to $tmf$

Let $tmf$ denote the connective spectrum of topological modular forms, and let $T_{tmf}\to tmf$ be a $2$-local characteristic morphism. A map of ring spectra is an epimorphism on homotopy groups when it induces a surjection on $\pi_*(-)$.

Epimorphism conjecture. A $2$-local characteristic morphism

$$T_{tmf}\longrightarrow tmf$$

induces an epimorphism on $\pi_*(-)$.

The conjecture is motivated by analogous characteristic morphisms to $kU$ and $kO$, which the source says induce epimorphisms on homotopy groups. No proof or counterexample is given for the proposed $tmf$ statement.