Quadratically hyponormal weighted shifts with recursive tail

We characterize positive quadratic hyponormality of the weighted shift W α ( x ) associated to the weight sequence α ( x ) : 1 , 1 , x , ( u , v , w ) ∧ with Stampfli recursive tail, and produce an interval in x with non-empty interior in the positive real line for quadratic hyponormality but not positive quadratic hyponormality for such a shift.