Does your client deal with video? If so, then it is likely that over the course of the next 5-10 years the HTML5 video tag could replace Flash as the default video player on the web. However, even if HTML5 chips away at Flash's relevance as a video player; why would that affect Flex's ability to provide a Smart Application GUI?
Does your client build games? It is possible--although I believe unlikely--that the HTML5 Canvas element will be used for a lot of web games. IF this aspect of HTML5 chips away at Flash's relevance as a web game platform, why would that affect Flex's ability to provide a smart Application GUI?
Additionally, why did your client choose Flex over an AJAX solution? Would the same reason apply to an HTML5 choice?
It is also probable that Adobe will keep innovating The Flash Player and Flex faster than HTML5. Flex has a mature toolset with lots of support.
I'd also ask why is it important for your client to provide applications support on mobile devices? The answer to that question will have a big affect on the client's decision whether to use Flash or not.
I don't have any specific numbers or case studies to share, though. YouTube had a Blog post about why they chose Flash. So did Hulu. Of course, I do not know if either of them are using Flex.