I refuse to wean a child from something their parent still allows IF it causes grief or unnecessary strife here.
If DCK is still allowed to have a Binky at home, of course he isn't going to sleep at your house.
I would continue to allow him to have it at nap time but would INSIST that the one mom provides is new and in good condition so that it isn't a choking hazard.
If she does not supply a new one or one in good condition, refuse to keep him.
If he still gets one at home and you are not allowing it, its going to be nothing but a continuous power struggle with NO winners and NO NAPPERS.
