In the book The unstable Adams spectral sequence for free iterated loop spaces, R.J. Wellington, Mem. Amer. Math. Soc. 258, 1982, p. 32

Question: When $p=2$, $k\geq 2$ and $n=\infty$, what is the dimension of each generator $j^*(x_I^*)$? I do not quite understand the argument.

The related context is in the following.