This answer links to Spaceflight 101's One Week to Jupiter – NASA’s Juno Spacecraft en-route to Gas Giant after five-year Journey which says
Also, Juno will put Einstein’s General Theory of Relativity to the test by studying orbital frame dragging close to a massive body.
Question: How, and How Well will Juno measure the effects of frame dragging? What instruments and/or equipment will be used to detect frame dragging? Will it be a fairly precise quantitative determination, or more of a qualitative "yep, there's an effect" verification?
The biggest effect of Jupiter's rotation is the gravitational quadrupole or J2 moment caused by Jupiter's oblateness, which is a result of the rotation. It causes the apses of Juno's orbit to precess substantially. See Juno's original orbit around Jupiter - is this apsidal precession? If so, need expression.
above: a plot of Juno's original planned orbit. Because of a slow responding helium check valve, Juno actually was left in its higher orbit, but still experiences precession of the apses. Plot from here.
So the frame dragging would have to be separated from all other effects due to deviations of Jupiter's gravity field from a pure monopole. This sounds extremely difficult!