Single-source capacitated facility location problems (SSCFLPs) are well known in the operations research literature. A set of facilities is opened and each customer is assigned to exactly one open facility so that the capacity at each facility is respected. This customer assignment, however, deprives customers from choosing facilities according to their individual preferences. If customers are allowed to deviate to their most preferred open facility within a solution for the SSCFLP, these re-assignments might lead to facilities serving more demand than they are capable of and, therefore, can render feasible solutions infeasible. Preference constraints take individual customer preferences into account before computing a solution and ensure that each customer is served at their most preferred open facility while respecting the capacity at each facility. This problem is called single-source capacitated facility location problem with customer preferences (SSCFLPCP) and is known to be strongly NP-hard as an extension of the SSCFLP, in which each customer is indifferent regarding all facilities. In this paper, we contribute new complexity results for the SSCFLPCP. We show that the type of customer preferences, e.g., geographically closest assignments or strict preferences, has a vital impact on the theoretical complexity of the considered instances. Afterwards, we focus on the special case in which preferences are defined by closest assignments and give several results for the SSCFLPCP which differ from complexity results known for the SSCFLP. Our findings provide a deeper understanding of the transition from polynomially solvable cases to strongly NP-hard cases of the computational complexity of the SSCFLPCP.