Thierry Coquand has a list of open problems, based on Vladimir Voevodsky's research on Dependent Type Theory. Scroll to the end of this <A HREF="https://web.archive.org/web/20190518164032/https://eutypes.cs.ru.nl/eutypes_pmwiki/uploads/Meetings/Coquand.pdf">presentation.</A>

And I presume you know the HoTT open problem list at <A HREF="I presume you know this list: https://ncatlab.org/nlab/show/open+problems+in+homotopy+type+theory">nLab</A>