 Hyperphantom map of spectra

A hyperphantom map of spectra is a stable map $p:D\to E$ such that $pu=0$ for every $u:{\Sigma }^{-j}{\Sigma }^{\infty }X\to D$, for arbitrary based spaces $X$ and all nonnegative $j$. Equivalently it induces zero on the entire represented cohomology theory. This is a stronger condition than being a phantom map of spectra, which tests only finite spectra.