Good question. I suspect they would, because for all their animosity with the radical Kurds, Kurdistan is Turkey’s largest non-European trading partner.
They might demand a price from the Kurds, but it might not be intolerable.
But if there is a big direct conflict shaping up between Iraq and the Iraqi Kurd Peshmurga Iraq probably won’t want to see more Kurds, and the Syrian Kurds might be jumping from the pan into the fire so maybe they think it is best to sit tight.