Many companies make use of customer service chats to help the customer and try to solve their problem. However, customer service data is confidential and as such, cannot easily be shared in the research community. This also implies that these data are rarely labeled, making it difficult to take advantage of it with machine learning methods. In this paper we present the first work on a customer’s problem status prediction and identification of problematic conversations. Given very small subsets of labeled textual conversations and unlabeled ones, we propose a semi-supervised framework dedicated to customer service data leveraging speaker role information to adapt the model to the domain and the task using a two-step process. Our framework, Task-Adaptive Fine-tuning, goes from predicting customer satisfaction to identifying the status of the customer’s problem, with the latter being the main objective of the multi-task setting. It outperforms recent inductive semi-supervised approaches on this novel task while only considering a relatively low number of parameters to train on during the final target task. We believe it can not only serve models dedicated to customer service but also to any other application making use of confidential conversational data where labeled sets are rare. Source code is available at https://github.com/gguibon/taft