The 520/523/543 use a conventional ring gear, and just so happen to have almost the exact same dimensions as the neon T350 diff, which is why the obx works in those transmissions.
The 555/568 on the other hand are a bit weird. In order to use 4 pinions, the carrier had to be solid all the way around. But that removes the normal opening into which you would install the spider/pinion gears in a regular diff. So what Chrysler did was to make not only the ring gear, but the entire bottom of the diff unbolt, exposing the inside of the carrier and allowing for the spiders and pinions to be installed.
Since the obx only fit's in a 520/523/543 because they share the same diff design there is no way to use it with the very unconventional 555/568 "ring gear". In addition, since the tooth count and pitch on the 555/568 mainshaft matches on the 555/568 "ring gear" you have to install the mainshaft that matches the ring gear you wish to use, in this case the one from a 520/523/543.
Now, if you want the lower gearing, you can use the 3.77's from a 92+ 523/543. They retain the conventional ring gear and are as close to the original 3.85's as is possible while still allowing for the use of an obx.