Generative AI (hereinafter GenAI) technology, such as ChatGPT, is already influencing the higher education sector. In this work, we focused on the impact of GenAI on the academic integrity of assessments within higher education institutions, as GenAI can be used to circumvent assessment approaches within the sector, compromising their quality. The purpose of our research was threefold: first, to determine the extent to which the use of GenAI can be detected via the marking and moderation process; second, to understand whether the presence of GenAI affects the marking process; and finally, to establish whether authentic assessments can safeguard academic integrity. We used a series of experiments in the context of two UK-based universities to examine these issues. Our findings indicate that markers, in general, are not able to distinguish assessments that have had GenAI input from assessments that did not, even though the presence of GenAI affects the way markers approach the marking process. Our findings also suggest that the level of authenticity in an assessment has no impact on the ability to safeguard against or detect GenAI usage in assessment creation. In conclusion, we suggest that current approaches to assessments in higher education are susceptible to GenAI manipulation and that the higher education sector cannot rely on authentic assessments alone to control the impact of GenAI on academic integrity. Thus, we recommend giving more critical attention to assessment design and placing more emphasis on assessments that rely on social experiential learning and are performative rather than output-based and asynchronously written.