Semicompeting risks and interval censoring are frequent in medical studies, for instance when a disease may be diagnosed only at times of visit and disease onset is in competition with death. To evaluate the ability of markers to predict disease onset in this context, estimators of discrimination measures must account for these two issues. In recent years, methods for estimating the time-dependent receiver operating characteristic curve and the associated area under the ROC curve have been extended to account for right censored data and competing risks. In this paper, we show how an approximation allows to use the inverse probability of censoring weighting estimator for semicompeting events with interval censored data. Then, using an illness-death model, we propose two model-based estimators allowing to rigorously handle these issues. The first estimator is fully model based whereas the second one only uses the model to impute missing observations due to censoring. A simulation study shows that the bias for inverse probability of censoring weighting remains modest and may be less than the one of the two parametric estimators when the model is misspecified. We finally recommend the nonparametric inverse probability of censoring weighting estimator as main analysis and the imputation estimator based on the illness-death model as sensitivity analysis.