Infant naps

[bright sparks]

Regardless of the cause, I think you have to treat this differently than the average transition and alter your expectations. This could take months and if the child is still rocked to sleep at home at the same time, I'd be inclined to give notice. You can't trust this parent and even if she said she was on board and doing exactly as you tell her, I doubt she would actually be honest and do it.
If you choose to carry on providing care then stick to a VERY STRICT timed routine for meals and naps. Naps will ultimately be CIO and this child should be separated from the rest of the group and preferably by a floor too. Do not go back into the room AT ALL for the duration of the nap. If it doesn't work even ever so slightly within a couple of weeks, like periods of quiet, difference in intensity of crying, then it's a fair bet that mum isn't doing the same at home and at that point I'd terminate. Apart from the lying, the child isn't a good fit for your daycare and you don't have a collaborative caregiving relationship with the parents.