Interesting logic, but it seems more likely it would have simply been destroyed. That's what happened to the Temple, after all.
The relics in the temple were expressly referenced in Daniel (?) -— the “writing on the wall episode” where the Babylonian King is using the gold/silver implements for a dinner party.
I would assume he got the while lot, including the Ark itself.
The Ark, IMHO, has been stripped of its gold, and now resides in people’s gold filings and jewelry around the world -— like most things gold -— they keep getting melted, combined and reused.
The real treasure — the tablets -— were cast aside and trod underfoot. I would not be suprised to find them used as bricks in some stone wall somewhere.