It maybe should not be a surprise that there is a symbiotic relationship between high inequality in a society and low social mobility. In a highly unequal society, many advantaged parents will do all they can to ensure that their children do not slip down the economic ladder--they know that it goes a long way down. And if incomes and wealth are unequally distributed, they have the resources to help them. At the same time, they may realise that higher rates of social mobility, in relative terms, cannot be a one-way street. If policy helps increase the chances of someone starting in a less privileged position to go up the social scale, that must mean that someone else's chance of going down has to rise, which may include their own children, and does not then seem so attractive. While many favour increased upward mobility, few want to mention the increased downward mobility that has to go with it (in terms of relative positions, at least).
wonder how such parents would feel about dismantling the ladder entirely