In group decision making problems, given the importance of obtaining an accepted solution by the whole group, the consensus has attained a great attention and it is virtually a major goal of these problems. Consensus, as traditionally meant to be a full and unanimous agreement, is often not reachable in practice. A more realistic approach is to use softer consensus measures, which assess the consensus degree in a more flexible way, and therefore reflect the large spectrum of possible partial agreements, guiding the consensus process until widespread agreement is achieved among experts. In particular, the interpretation of the consensus based on the concept of fuzzy majority has been used in the most of the consensus models proposed in the literature, as it is more human-consistent and suitable for reflecting human perceptions of the meaning of consensus. However, there are still some open questions to be addressed. In this paper, we are going to highlight some issues in order to focus researcher's attention on new problems that arise when using consensus models based on soft consensus measures in real-world applications.