Iraqi Interior Ministry accuses ISIL of Stealing Antiquities from Iraqi Museums

The ISIL state (Tikrit and Mosul marked by me,
plus one possible smuggling route suggested)

The Iraqi Interior Ministry  has accused the "Islamic State of Iraq and the Levant" (ISIL) of stealing Iraqi antiquities from Mosul and Tikrit museums and other archaeological sites ('ISIL steals, smuggles Iraqi antiquities: ministry', 2014-07-24):

"ISIL gangs have seized at least 100 artefacts, including statues, jewellery, [pott]ery and ancient historical items dating back to the Babylonian, Sumerian, Akkadian and Abbasid eras, and smuggled them to Syria to sell them there through organised international gangs," Col. Mahmoud al-Issawi, director of the ministry's antiquities protection unit, told Al-Shorfa. "These barbaric acts are a proof that this group which makes claims to a caliphate and Islam is nothing but an organised gang that steals, loots, launches armed robberies and kills citizens," he said. The authorities have notified neighbouring and regional countries about the stolen items, al-Issawi said. [...].

There is however the interesting problem that somewhere in Iraq there possibly are private warehouses full of the stuff claimed looted from sites from 2003 (and perhaps before) much of which seems not to have hit the international market, and not all of which seems likely to have been smuggled out yet (for if so, where and why?). So why are ISIS not seizing these? Do these caches exist?