Minimum folded ribbonlength conjecture for Möbius-band unknots with linking number ±3

Let $\mathcal{U}_{w,F}$ be a folded ribbon unknot that is a Möbius band, with ribbon linking number ${\operatorname{Lk}}(\mathcal{U}_{w,F})=\pm3$. Let ${\operatorname{Rib}}$ denote folded ribbonlength. Minimum folded ribbonlength conjecture. The minimum is

$${\operatorname{Rib}}([\mathcal{U}_{w,F}])=\inf_{\substack{\mathcal{U}_{w,F}\in\operatorname{Möbius},\\ {\operatorname{Lk}}(\mathcal{U}_{w,F})=\pm3}}{\operatorname{Rib}}(\mathcal{U}_{w,F})=3\sqrt{3}.$$

The value is attained by the equilateral-triangle example established earlier for folded ribbon $3$-stick unknots. The conjecture asks whether allowing arbitrary Möbius-band folded ribbon unknots can lower this value.