In this work, we focus on some conditional extreme risk measures estimation for elliptical random vectors. In a previous paper, we proposed a methodology to approximate extreme quantiles, based on two extremal parameters. We thus propose some estimators for these parameters, and study their asymptotic properties in the case of heavy-tailed distributions. Thereafter, from these parameters, we construct extreme conditional quantiles estimators, and give their consistency properties. Using recent results on the asymptotic relationship between quantiles and other risk measures, we deduce estimators for extreme conditional Lp-quantiles and Haezendonck-Goovaerts risk measures. In order to test the efficiency of our estimators, we propose a simulation study. A financial data example is also proposed.