Before revision:  In past work on this and also the more general mixture-of-products problem (X_i independent conditional on U, but not necessarily iid), a barrier at m^{O(k^2) a sample of size \exp(k\log k) (with m=2k). It is known that the sample complexity of any solution to the identification problem must be \exp(\Omega(k)) \left(}\right)\left(\right) and post-sampling runtime of only O(k^{2+o(1)}) arithmetic operations.
After revision:  In past work on this and also the more general mixture-of-products problem (X_i independent conditional on U, but not necessarily iid), a barrier at m^{O(k^2) a sample of size \left(}\right)\left(\right) and post-sampling runtime of only O(k^{2+o(1)}) arithmetic operations.