You might be able to slave the ring to a value present in the morph target. In short, just using my own figurin', the MT you're using only applies to that one group. It doesn't cover the extra "ring" bone's group. So, the ring "doesn't know what to do" other than follow the parent->child relationship of the bone structure. As far as the ring is concerned, nothing has happened at all when your custom MT value changes. But, you could create a movement morph for that ring group that was dependent upon the morph target's value in the head group. So, let's say you have an MT in the head group for "open mouth." When that value reaches xx, your ring would move xx.

PS - Conforming items that closely conform to the lower face are... difficult. That's because there's no group for them to follow and much of V4's mouth movements are just MTs, which don't effect groups that they don't span to. You can, however, do some thing with "superconforming" items to get custom-created MTs to follow certain values in their corresponding conforming parent. That's another avenue you can try for general purposes, but it won't work for custom rigging/groups, since those have no analogue in the parent object.