It makes the problem worse because the object picks up a huge black reflection from the lens AND black cloth. Just to highlight again this doesn't represent a typical product shoot scenario because of the requirement to shoot straight on at the subject.
I'm not sure if this would help, since the subjects are of similar properties (mirror-like), but I was once speaking with a watch collector and he mentioned that the experienced watch photographer who shot some of his watches shrouded his lens with a very large piece of white paper/cardboard and surrounded the entire area just in front of the watch with white (since the watch picks everything up. The key, I'm guessing, is LARGE paper/cloth/cardboard and placement. And carefully selected spectral reflections. If not, the jewelery looks flat and dead.
If white is too glaring, try different shades of white/very light cream/grey, then adjust the WB in post.
Another way is using a light cone.
There are other ways, but it gets really technical.
Even then, I suspect a lot of post work is still required to perfect and enhance the shot. It all depends on what degree of quality you want/need.
A tilt shift lens, while the logical line of thought, would probably not have enough movement. Besides, go more than 9mm and the IQ usually drops like a rock. To avoid the reflection of the lens/camera, you would probably need to shift much more than 9mm. Which is why high-end watch work is usually done with a view camera.