They have a precious item that is there, but it is not the true Ark.
Solomon had an item made to give Menelik to take back to Ethiopia. This item was to be acknowledgement of his relationship and of the faith of the Israelites. Solomon later had drifted from stern adherence to obedience to God through his fascination with pagan wives. The creation of the Ark like but not Ark actual object for Menelik was the beginning of his drifting from solid theology and obedience to God. So the object was Solomon's creation as a type of copy to give to this son of his as an object of veneration and a way to "evangelize" the faith of the Israelites. The Qur'an even has a mention of Solomon doing metallurgy crafts that tapped not so holy sources, so Solomon veered from wisdom even as "trying" to still be the consecrated King of the Israelites. He therefore saw nothing wrong with making an ark like copy to give to his son to take back to Ethiopia and that is what exists to this day. The true Ark would never have been allowed to leave, either by Solomon or by God. The Ark left only when Israel fell.