The emergence of generative artificial intelligence (GenAI) text generator tools and the potential challenges for higher education (HE) have characterised informal academic discussion on multiple fora. Specifically examining the case of English medium instruction (EMI) assessment academic integrity, this study sought to explore this conundrum by conceptualising threats and possible recommendations to counter these by creating a problem-solution matrix for key stakeholders considering the scarce academic literature available. An exploratory Delphi technique was employed as a way of generating ideas, gauging expert perspectives, and establishing consensus based on the premise of wisdom-of-(expert)-crowds. In the data collection stage, this new use of the mixed-methods methodology in the field included iterative Delphi questionnaire rounds and concurrent focus group sessions with a panel of 26 international experts. Quantitative and qualitative data were analysed using descriptive statistics and thematic analysis, respectively. The resulting GenAI and EMI Assessment Problem-Solution Matrix is an empirically informed instrument for key stakeholders in EMI HE that exemplifies a range of GenAI-induced issues and recommendations as to how to proceed going forward in EMI HE pedagogical settings. This contributes to the field in line with broader theoretical assessment principles, particularly with those seeking to mitigate inequitable practices. Further contextual matters pertaining to social justice were highlighted, such as the effects of the massification and commodification of HE on the role of assessment in both EMI didactic contexts and others. The findings here take a step towards addressing the gaps identified but also represent a means of sparking much-needed further discussion in both extant literature and praxis.