  Let D be a piecewise smooth bounded convex Reinhardt domain in C^2. Assume
that the symbols f and g are continuous on the closure of D and harmonic on the
disks in the boundary of D. We show that if the product of Hankel operators
H^*_f H_g is compact on the Bergman space of D, then on any disk in the
boundary of D, either f or g is holomorphic.
