I suppose that plays a role in the proof at some point. But they emphasize that the statement does not follow from abstract properties of anything. They emphasize that the proof uses the hard computations of $K(n)$-(co)-homology of Eilenberg-MacLane spaces in

Ravenel, D. and S. Wilson. The Morava K-theories of Eilenberg-Mac Lane spaces and the Conner-Floyd conjecture. Amer. J. Math. 102 (1980), no. 4, 691748.

Johnson, D. and S. Wilson. The Brown-Peterson homology of elementary p-groups. Amer. J. Math. 107 (1985), no. 2, 427453.

So properties there are related to the $K(n)$ being ’field spectra’ (p. 18 of notes)?

]]>am starting *K(n)-local stable homotopy theory* .