Unit-group equality for powers of two

Let $k$ be the ground field, let $T$ be the coordinate ring considered in the paper, let $T^\ast$ be its group of units, and let $n$ be the relevant degree or group order. Let $G$ act on the class groups $\operatorname{Cl}(S_i)$. Power-of-two unit-group conjecture. If

$$n=2^s,\qquad s>0,$$

then

$$T^\ast=k^\ast.$$

The statement is intended to extend the argument proving the case $n=4$. Iterating that argument requires knowing that the elements of order two in $\operatorname{Cl}(S_i)$ are fixed by $G$, and the supplied text does not establish this in general.