Much of the cost and effort required during the software testing process is invested in performing test maintenance - the addition, removal, or modification of test cases to keep the test suite in sync with the system-under-test or to otherwise improve its quality. Tool support could reduce the cost - and improve the quality - of test maintenance by automating aspects of the process or by providing guidance and support to developers. In this study, we explore the capabilities and applications of large language models (LLMs) - complex machine learning models adapted to textual analysis - to support test maintenance. We conducted a case study at Ericsson AB where we explore the triggers that indicate the need for test maintenance, the actions that LLMs can take, and the considerations that must be made when deploying LLMs in an industrial setting. We also propose and demonstrate a multi-agent architecture that can predict which tests require maintenance following a change to the source code. Collectively, these contributions advance our theoretical and practical understanding of how LLMs can be deployed to benefit industrial test maintenance processes.