Indeed. Always check the sources! That's not to say it can't be true - but I suggest looking for the real quotes from Tolkien himself which would support this. In general, like said above, I would not expect the annexation to happen. Maybe looking into the Appendices could bring some light... that's what I would expect. I looked into them only briefly and I didn't find anything in the part about Rohan (Appendix A II), only this, which would actually point to the contrary:
Quote:
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In Gondor the King Elessar now ruled, and in Arnor also. In all the lands of those realms of old he was king, save in Rohan only; for he renewed to Éomer the gift of Cirion, and Éomer took again the Oath of Eorl.
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So, here we go. "Rohan only" means, well, Rohan only
and Enedwaith
would be the part of the "reams of old", at least in the times of greatest might of the heirs of Elendil.