The hidden-extension conjecture in the tmf-ASS

Let $x_i=h_{i,0}^2$ in the $\mathrm{tmf}$-Adams spectral sequence for $Z$. Hidden-extension conjecture. There are hidden extensions

$$v_2^{2^{i+1}}x_i=v_2x_{i+1}^2.$$

This is suggested by comparing the known $d_1(h_{i+2,1})=v_2^{2^{i+1}}x_i$ with the conjectural Adams differential $d_4(h_{i+2,1})=v_2x_{i+1}^2$; it would assemble the indicated torsion families into $v_2$-periodic families in $\pi_*Z$.