This excellent answer has me wondering if there are any measurements or estimates of gravitational multipole moments of Jupiter or Saturn beyond J2.
Body J2 ×1E+06 ------- --------- Earth 1,082.63 Jupiter 14,736 Saturn 16,298.
Question: Have gravitation multipoles of Jupiter and Saturn beyond J2 been measured or at least estimated? At least the zonal harmonics perhaps? For estimates, please distinguish observational estimates based on natural or artificial satellite orbits from those based on planetary oscillation observations or planetary dynamical predictions.