When I have worked in the past with custom carrier boards for the TX2 and the HDMI output is not working then it was due to lack of support in the device tree (i.e you would need a patch for the device tree). So I would say this is most likely your case as well. Is the carrier board your own design or did you purchased it from some hardware manufacturer? This has been a common issue in the past but to provide more guidance you would need to share more information of your design or ask for support from the carrier board manufacturer.
Also, I am sure that if you browse through the forums your will find some threads with useful information of similar cases in the past, let me share some links of threads that I consider with a similar issue as this one: