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From the body of the question it seems like you might be interested in running a version of chromatic homotopy theory with n-dim FGLS, and it seems like the first step there would be to find a replacement for CP^infty, namely a (commutative?)-group-up-to-homotopy G such that H^*(G) is Z[x1,...,xn]...but from the title it seems like you're interested in whether n-dim FGLs play any role in the usual, 1-dim FGL-based chromatic story?
@pupshaw great point - hyperphantom maps were kind of how I came to this question anyway. Also, I'm not really married to my connectivity condition. I'll edit the question to remove that condition, then you should turn your comment into an answer!
