This doesn't explain why the relationship holds true even in countries with free or heavily subsidized daycare. Or why only daughters are affected, along many variables.
Even when childcare is heavily subsidised it use will still be dominated by the middle classes. If you are unemployed why go to the effort to cart the children you love down to some strangers even if you don't have to pay anything.
You were talking about low wage mothers, I responded to that. They are anyway more numerous than unemployed-for-reasons-unrelated-to-parenting mothers. (This can also make financial sense to subsidize, as otherwise they'll have a larger probability to go unemployed and permanently fall out of workfroce later when kids go to school)