The proposed plan is a sound one, providing that the the grout is injected underneath the active clay zone. This will allow the grout to compact the loose soils below the clay layer. Grout and clay do not mix.
The real question here is what is actually moving the property, the loose soils at 20 feet or the active clay layer at 1-7 feet. While the grout will take care of the problem at the deeper level, if the actual problem is at the shallow level, the grouting will not solve the problem.
Piers will solve the problem of the active soils by transfering the weight of the structure to the deeper soils hence bypassing the shrink/swell activity of the surficial clays.
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