This study is in the context of previous assumptions of a U-curved relationship: low equality == high birth rates, medium equality == low birth rates, high equality == high birth rates. That study shows that the last part doesn't seem to be true.
Poverty can probably actually lower birth rates by forcing women into the workforce. If a parent is at home, the marginal cost is a lot lower for another child than it is if people are paying for childcare.
It's pretty well-established that it isn't poverty per se but gender equality that predicts birth rates.