Rigidity conjecture for optimal paper Möbius bands

Let a paper Möbius band be a smooth isometric embedding of the flat Möbius band $M_{\lambda}$ in $\mathbb{R}^3$, and consider a sequence of smooth embedded paper Möbius bands whose aspect ratios converge to $\sqrt{3}$. Rigidity conjecture. The sequence converges, in the Hausdorff metric and up to isometries, to an equilateral triangle of semi-perimeter $\sqrt{3}$. This is the rigidity statement accompanying the optimality conjecture in the source; no resolution is supplied, so it remains open.