Adams's cofibration-generation conjecture for the thick subcategory C_2

Let $p$ be an odd prime, let $\mathcal C_2$ be the thick subcategory of finite $p$-local spectra of type at least $2$, and let $V(1)$ denote the Smith–Toda complex. Adams's conjecture. The thick subcategory $\mathcal C_2$ can be generated from $V(1)$ by iterated cofiberings. This is equivalent to $\operatorname{Ker}(\phi_2)=0$, or equivalently to $K_0(\mathcal C_2)\cong\mathbb Z$. The conjecture concerns generation without allowing retractions and would determine the Grothendieck group of $\mathcal C_2$; the source does not give a resolution.