Adams's refined generation conjecture for type-2 atomic spectra

Let $p$ be an odd prime, let $V(1)$ be the Smith–Toda complex, and let $X$ be a type-$2$ atomic spectrum. For a finite $p$-torsion atomic spectrum $X$, define $e(X)=k$ when $[X,X]/\operatorname{rad}\cong\mathbb F_{p^k}$. Adams's refined conjecture. At odd primes, $V(1)$ generates, by iterated cofiberings, every type-$2$ atomic spectrum $X$ such that $e(X)=1$. This is a refinement of Adams's generation conjecture after Xu's theorem, which shows that type-$n$ atomic spectra with bounded $e(-)$ generate $\mathcal C_n$. The refined assertion remains unresolved in the source.