Local Volatility smile dynamics

In The Local Volatility Surface (2008), Emanuel Derman analyzes the dynamics of the local volatility smile. He observes that, in a negatively skewed market, the evolution of the index is characterized by lower local volatility as the index level increases and higher local volatility as the index level decreases. This behavior causes the local volatility smile to shift in the opposite direction of the underlying asset's price movements. To illustrate this phenomenon, Derman includes the following