Population size estimation has been a problem of interest in many areas like epidemiology, ecology, demographic study. This task is executed by a sampling procedure called capture-recapture which collects multiple samples or lists from the population of interest with possible overlaps. Existing approaches require assumptions on the collected lists to ensure identifiability. We have previously considered conditional independence assumption between two lists and developed a non-parametric doubly-robust estimator under that assumption. The independence assumption may be too restrictive in many cases where it is impossible to avoid interaction among different sources, for example, patient list collected from medical centres. We consider a more general setting by allowing dependence between the lists. Relaxing the independence assumption compromises identifiability of the parameter of interest. The parameter is only partially identifiable after this relaxation. A point estimator may not be feasible anymore. We extend our previous estimator to obtain bounds of the total population size with robust properties. This estimator is applied to Peru Internal Conflict data from 1980.