Localizing the Gardam unit: the support geometry of units in F_2[P] and its non-unique-product relatives

Gardam's counterexample to the Kaplansky unit conjecture is a unit of $\mathbb{F}_2[P]$, $P$ the Promislow group, whose support has $21$ elements, and Gardam has asked whether $21$ is least possible. Over $\mathbb{F}_2$ the unit equation is a parity condition on a pair of supports, and for amenable