Seems unlikely, considering how many chances there were for it to be taken elsewhere after Rome was sacked by Vandals.
Still, the Vatican does have underground storage said to be square miles in size, so who knows.
The Vandals may have taken the items from Jerusalem to Carthage in 455. If so, they probably ended up in Constantinople when Justinian’s army conquered the Vandal kingdom. Whether they were there in 1204, I don’t know—perhaps some Crusader made off with it when Constantinople was captured in the Fourth Crusade. Or the Venetians could have grabbed it.