We prove that for any non-trivial product-type action a of SUq(n) (0 < q < 1) on an ITPFI factor N, the relative commutant (N-alpha)' boolean AND N is isomorphic to the algebra L-infinity(SUq(n)/Tn-1) of bounded measurable functions on the quantum flag manifold SUq(n)/ Tn-1. This is equivalent to the computation of the Poisson boundary of the dual discrete quantum group <(SUq(n))over cap>. The proof relies on a connection between the Poisson integral and the Berezin transform. Our main technical result says that a sequence of Berezin transforms defined by a random walk on the dominant weights of SU(n) converges to the identity on the quantum flag manifold.