Annular folded ribbonlength conjecture for unknots with prescribed linking number

Let $n$ be a nonnegative integer, and let $\mathcal{U}_{w,F}$ be a folded ribbon unknot that is a topological annulus, with ribbon linking number ${\operatorname{Lk}}(\mathcal{U}_{w,F})=\pm n$. Let ${\operatorname{Rib}}$ denote folded ribbonlength. Annular folded ribbonlength conjecture. The minimum folded ribbonlength is

$${\operatorname{Rib}}([\mathcal{U}_{w,F}])=2n.$$

The theorem preceding this conjecture proves the value $2n$ when the writhe is zero. The conjecture asserts that allowing nonzero writhe cannot produce a smaller folded ribbonlength; folds and crossings each contribute at least $2$ in the relevant lower-bound argument.