The parabola conjecture for the telescope of $Z$

Let $\widehat{Z}$ be the telescope of the $v_2$-self-map on $Z$, and let $x_i:=h_{i,0}^2$. Parabola conjecture. The localized Adams spectral sequence for $Z$ collapses at $E_5$, and

$$\pi_*\widehat{Z}\cong\mathbb{F}_2[v_2^{\pm}]\otimes E[\widetilde{h}_{2,1},h_{3,0},h_{3,1},x_3,x_4,x_5,\ldots].$$

Moreover, the telescope conjecture is false: the kernel of $\pi_*\widehat{Z}\to\pi_*Z_{E(2)}$ is $(x_3,x_4,\ldots)$, while $(\widetilde{h}_{4,1})\subset\pi_*Z_{E(2)}$ maps isomorphically onto the cokernel. This is the paper's proposed anti-telescope description for $Z$.