For an unknown reason, Vanilla2 does not allow HTML in Inbox conversations. As far as I can tell, this issue has existed since the release of Vanilla2 last year. This bug has remained at least until the latest revision of Vanilla - 126.96.36.199. This solution can be easily fixed by modifying one line:
On or around Line 36, change
After the change, HTML will render properly in Inbox Conversations.