Within the minimal supersymmetric extension of the Standard Model, the best fit to the most recent precision-measurement data requires charginos and neutralinos, with dominant Higgsino components and with masses within the reach of LEP1.5 ( S =140 GeV ). In this work, we present a detailed analysis of the neutralino and chargino production processes for the favoured region of parameter space, that is low values of |μ| and either low or large values of tan β. We find that chargino and neutralino searches can cover the Higgsino region in the (μ, M 2) plane for values of M 2 ≲ 1 TeV, at the next phases of the LEP collider. We also show that, due mainly to phase-space constraints, the lightest neutralinos should be more easily detectable than charginos in most of the parameter space preferred by precision-measurement data.