We present a result from our work with Michael Eichmair and Abraão Mendes concerning initial data rigidity results (CMP, 2021), and consider some consequences. In a note with Piotr Chruściel (CQG 2021), we showed how to use this result, together with arguments from Chruściel and Delay's proof of the their hyperbolic PMT result, to obtain a hyperbolic PMT result with boundary. This, in particular, will be discussed.