  We describe the modulo $2$ de Rham-Witt complex of a field of characteristic
$2$, in terms of the powers of the augmentation ideal of the
$\mathbb{Z}/2$-geometric fixed points of real topological restriction homology
TRR. This is analogous to the conjecture of Milnor, proved by Kato for fields
of characteristic $2$, which describes the modulo $2$ Milnor K-theory in terms
of the powers of the augmentation ideal of the Witt group of symmetric forms.
Our proof provides a somewhat explicit description of these objects, as well as
a calculation of the homotopy groups of the geometric fixed points of TRR and
of real topological cyclic homology, for all fields.
