Isn't the relationship in the graph simply the result of how income inequality and social mobility are defined? The lower the income inequality the less you need to earn (relative to the baseline income) to move up the social ladder (defined in terms of income).

It is usually a strong welfare states that helps children of lower classes achieve the same as those of upper classes.

But the strong welfare state reduces the spread between lower and upper classes and reduces the delta required to be "mobile".