In the life-cycle of software development, testing plays a crucial role in quality assurance. Proper testing not only increases code coverage and prevents regressions but it can also ensure that any potential vulnerabilities in the software are identified and effectively fixed. However, creating such tests is a complex, resource-consuming manual process. To help developers and security experts, this paper explores the automatic unit test generation capability of one of the most widely used large language models, GPT-4, from the perspective of vulnerabilities. We examine a subset of the VUL4J dataset containing real vulnerabilities and their corresponding fixes to determine whether GPT-4 can generate syntactically and/or semantically correct unit tests based on the code before and after the fixes as evidence of vulnerability mitigation. We focus on the impact of code contexts, the effectiveness of GPT-4's self-correction ability, and the subjective usability of the generated test cases. Our results indicate that GPT-4 can generate syntactically correct test cases 66.5\% of the time without domain-specific pre-training. Although the semantic correctness of the fixes could be automatically validated in only 7. 5\% of the cases, our subjective evaluation shows that GPT-4 generally produces test templates that can be further developed into fully functional vulnerability-witnessing tests with relatively minimal manual effort. Therefore, despite the limited data, our initial findings suggest that GPT-4 can be effectively used in the generation of vulnerability-witnessing tests. It may not operate entirely autonomously, but it certainly plays a significant role in a partially automated process.