Lunar ranging and the Cassini measurement already have tested the part of relativity governing the geodetic effect at comparable levels of precision.
In fact, the international group of radio astronomers who discovered J0737-3039 has already witnessed at least four general-relativity effects, and possibly a fifth (the geodetic effect), resulting in the theory's most precise test in a strong gravitational environment.
The other is the "geodetic effect," the curvature of space-time due to the presence of mass, which is already well known but will be measured more accurately than before--in hopes that new physics beyond relativity might start showing up.
In most situations the geodetic effect is extremely small, but it is enough to explain every known property of gravity with perfect precision--so far.
Known as the geodetic effect, the tilt would amount to 6.6 arc-seconds a year, or 360[degrees] in 200,000 years.
In a polar orbit the tilt due to the geodetic effect would be at a right angle to the tilt caused by the gravitomagnetic effect, making both effects detectable.