Halpern–Weaver optimality conjecture for paper Möbius bands

Let $M_{\lambda}=([0,1]\times[0,\lambda])/\sim$ be the flat Möbius band, where $(x,0)\sim(1-x,\lambda)$, and let a paper Möbius band be a smooth isometric embedding $I:M_{\lambda}\to\mathbb{R}^3$. Let $\lambda_0$ be the infimal aspect ratio $\lambda$ for which a smooth embedded paper Möbius band exists. Halpern–Weaver's optimality conjecture. A smooth embedded paper Möbius band has aspect ratio greater than $\sqrt{3}$. Hence

$$\lambda_0=\sqrt{3}.$$

Halpern and Weaver proved that $\lambda_0\in[\pi/2,\sqrt{3}]$; the conjecture asserts optimality of the upper bound and remains unresolved in the supplied source.