  It has been conjectured by Ambrosanio, Kane, Kribs, Martin and Mrenna (AKM)
that the CDF event $p \bar p \to e^+ e^- \gamma \gamma + missing E_T$ is due to
a decay chain involving two neutralino states (the lightest and the
next-to-lightest ones). The lightest neutralino ($\chi_{AKM}$) has been further
considered by Kane and Wells as a candidate for cold dark matter. In this paper
we examine the properties of relic $\chi_{AKM}$'s in their full parameter
space, and examine the perspectives for detection by comparing theoretical
predictions to sensitivities of various experimental searches. We find that for
most regions of the parameter space the detectability of a relic $\chi_{AKM}$
would require quite substantial improvements in current experimental
sensitivities. The measurements of neutrino fluxes from the center of the Earth
and of an excess of $\bar{p}/p$ in cosmic rays are shown to offer some
favorable perspectives for investigating a region of the $\chi_{AKM}$ parameter
space around the maximal $\tan \beta$ value allowed by the model.
