The Letter of the Prophet (pbuh) displayed in the National Museum of Oman in Muscat, which opened its doors in 2016, is one of several documents claining to have been written during the lifetime of the Prophet Muhammad (c. 570 CE in Mecca - 632 CE in Medina).
The label at the showcase in the museum gives the following information :
Letter of the Prophet Muhammad,
to Abd and Jaifar, joint Kings of Oman (facsimile),
Ink on leather,
8 AH/ 630 CE.
It clearly states that the document on display is a faksimile (a one to one copy) but doesn't say where the original is kept.
For more information see the article by LAKSHMI KOTHANETH in the Oman Observer from May 8, 2019 - link below