For each prime $p\geq 3$ let $\alpha\_p:S^{2p}\to S^3$ denote a representative of $\pi\_{2p}S^3$ of order $p$. Berstein and Hilton showed that for each $p$ the homotopy cofiber $C\_{\alpha\_p}$ of $\alpha\_p$ is a co-H-space which does not have the homotopy type of a suspension space. 


The maps $\alpha\_p$ give rise to a map $\alpha:\bigvee\_{p\geq 3} S^{2p}\to S^3$ whose cofiber $C\_\alpha$ is a co-H-space. Is it known whether $C\_\alpha$ has the homotopy type of a suspension space? 



Is it known whether this space is a suspension space?
Isn't the localization of a 2-connected suspension also a suspension? Then this cannot be a suspension. (Brayton Gray pointed this out to me.)


