The minimality conjecture for the Bousfield classes of A(n)

For each $n$, let $A(n)$ be the fiber of the natural map $T(n)\to L_{K(n)}T(n)$, and let $E$ be a spectrum with a finite acyclic, meaning that $E\wedge X=0$ for some nontrivial finite spectrum $X$.

The minimality conjecture for A(n). If $n\geq 2$, then $\langle A(n)\rangle$ is a minimal nonzero Bousfield class. Moreover, every such $E$ is Bousfield equivalent to a finite wedge of spectra $K(n)$ and $A(n)$; in particular,

$$\langle E\rangle=\bigvee_{\{n\mid E\wedge K(n)\neq 0\}}\langle K(n)\rangle\vee\bigvee_{\{n\mid E\wedge A(n)\neq 0\}}\langle A(n)\rangle.$$

This is proposed as a replacement for the false telescope conjecture. The source provides no resolution.