The v_2-local injectivity conjecture for the algebraic tmf resolution

Let $\mathrm{tmf}$ be the connective topological modular forms spectrum, let $\overline{\mathrm{tmf}}$ denote its cofiber in the unit decomposition, and let $\prescript{\mathrm{ass}}{}{E}^{s,t}_2(-)$ denote the mod $2$ Adams spectral sequence. For the sphere spectrum, consider the algebraic $\mathrm{tmf}$-resolution terms

$$\prescript{\mathrm{ass}}{}{E}^{s,t}_2(\mathrm{tmf} \wedge \overline{\mathrm{tmf}}^{n}).$$

Here $v_2^{-1}(-)$ denotes localization at the chromatic periodicity element $v_2$. The v_2-local injectivity conjecture. The map

$$\prescript{\mathrm{ass}}{}{E}^{s,t}_2(\mathrm{tmf} \wedge \overline{\mathrm{tmf}}^{n}) \longrightarrow v_2^{-1}\prescript{\mathrm{ass}}{}{E}^{s,t}_2(\mathrm{tmf} \wedge \overline{\mathrm{tmf}}^{n})$$

is injective for $s>0$. This would make the algebraic $\mathrm{tmf}$-resolution more accessible through its $v_2$-localization, analogous to corresponding results for the $\mathrm{bo}$- and $BP\langle 2\rangle$-resolutions; the source presents it as a proposed conjecture, and no resolution is supplied here.