Positive Laurent formula conjecture for the stacked twist map

Let $V$ and $W$ be matrices whose stacked matrix is $U:=\operatorname{stack}(V,W)\in\operatorname{Mat}^{\circ}(k+\ell,n)$, and let the stacked twist map satisfy $\theta(V,W)=(\widetilde W,\widetilde V)$. Write $\widetilde U:=\tau(U)=\operatorname{stack}(\widetilde W,\widetilde V)$, $Z:=W^\perp$, and $\widetilde Z:=\widetilde W^\perp$. Positive Laurent formula conjecture. There exist polynomials $H$ and $H'$ in the maximal minors of $V$ and $Z$, with nonnegative integer coefficients, such that for every $I\in{[n]\choose \ell+m}$ and $I\in{[n]\choose \ell}$ respectively,

$$\Delta_I(\widetilde Z)=\frac{H}{\pm\prod_{j\in I}\Delta_{[j-\ell,j+k)}(U)},$$

and

$$\Delta_I(\widetilde V)=\frac{H'}{\pm\prod_{j\in I}\Delta_{[j-\ell,j+k)}(U)}.$$

This predicts positivity and Laurent-type expressions for twisted Plücker coordinates; the source gives no resolution.