The Brown-Comenetz dualizing spectrum $I_{\mathbf{Q/Z}}$ is not detected by very many spectra: it is $BP, \mathbf{Z}, \mathbf{F}_2, X(n)$ for $n\geq 1$, and even $I_{\mathbf{Q/Z}}$-acyclic. However, if $X$ is any nontrivial finite spectrum, then $X\wedge I_{\mathbf{Q/Z}}$ is not contractible. This motivates a natural question: let $E$ be any spectrum such that $E\wedge I_{\mathbf{Q/Z}}$ is not contractible. Then, is it true that $\langle E\rangle \geq \langle X\rangle$ for some nontrivial finite spectrum $X$?

The Brown-Comenetz dualizing spectrum $I_{\mathbf{Q/Z}}$ is not detected by very many spectra: it is $BP, \mathbf{Z}, \mathbf{F}_2, X(n)$ for $n\geq 2$, and even $I_{\mathbf{Q/Z}}$-acyclic. However, if $X$ is any nontrivial finite spectrum, then $X\wedge I_{\mathbf{Q/Z}}$ is not contractible. This motivates a natural question: let $E$ be any spectrum such that $E\wedge I_{\mathbf{Q/Z}}$ is not contractible. Then, is it true that $\langle E\rangle \geq \langle X\rangle$ for some nontrivial finite spectrum $X$?