How do conditional moments of stable vectors depend on the spectral measure?

Let (X1,X2) be an [alpha]-stable random vector with 0 < [alpha] < 2, not necessarily symmetric. Its distribution is characterized by a finite measure [Gamma] on the unit circle called the spectral measure. It is known that if [Gamma] satisfies some integrability condition then the conditional moment E[[short parallel]X2[short parallel]p[short parallel]X1] can exist for some values of p greater than [alpha]. This paper provides a sufficient condition on [Gamma] for the existence of the conditional moment E[[short parallel]X2[short parallel]p[short parallel]X1] involving the maximal range of possible p's, namely p < 2[alpha] + 1.