I don’t think the rings would be close enough for easily detectable effects, but in terms of Jupiter in general, the answer is yes, and Juno is on the job. The

I don’t think the rings would be close enough for easily detectable effects, but in terms of Jupiter in general, the answer is yes, and Juno is on the job. The Juno spacecraft's suite of science instruments will primarily characterize and explore the three-dimensional structure of Jupiter's polar magnetosphere, auroras and mass composition.[4] As Juno is a Polar Orbit mission, it will be possible to measure the orbital frame-dragging, known also as Lense–Thirring precession, caused by the angular momentum of Jupiter.[5] https://en.m.wikipedia.org/wiki/Frame-dragging#Experimental_tests https://en.m.wikipedia.org/wiki/Frame-dragging#Experimental_...