Possibly they were put the Nea Church. Then we don't know what happened to them.
Possibly they were put into the Church of the Holy Sepulcher (Holy Sepulchre). Then, again, we lose track of them. Sean Kingsley thinks they were moved from there to the Greek Orthodox Monastery of Theodosius at Deir Dosi on the West Bank by the locum tenens Patriarch Modestus, just before the Persian invasion. There are a lot of caves in that vicinity, so our imaginations are free to run wild from there.
But Carl mentions a third possibility that I don't remember hearing before. Steven Fine has found a (granted, very late) tradition that the treasures were never taken from Rome!
I would be inclined to follow Procopius, who wrote much closer to the events he recounts. That would get the treasures back to Jerusalem and then who knows where.
The earlier posts in Carl's series on the Temple treasures are here and links. And for other PaleoJudaica links on the subject, start here.
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