Quadratically hyponormal weighted shifts with recursive tail

We characterize positive quadratic hyponormality of the weighted shift Wα(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.