Conjecture on units of open positroid coordinate rings

Let $\widetilde \Pi^{\circ}_\pi$ be the relevant open positroid variety, and let $\mathbb{P}_\pi$ be the free abelian group of Laurent monomials in the target frozen variables. The group of units of $\mathbb{C}[\widetilde \Pi^{\circ}_\pi]$ consists of invertible regular functions on this coordinate ring. Units conjecture. The group of units of the algebra $\mathbb{C}[\widetilde \Pi^{\circ}_\pi]$ coincides with the group $\mathbb{P}_\pi$. This assertion identifies all units of the open positroid coordinate ring with Laurent monomials in the target frozen variables; the supplied text gives no resolution, so its status is open.