Conjecture on the structure of a minimal configuration in Moser's worm problem

Let $\mathcal{L}$ be the fixed line segment, $\mathcal{S}$ the square, and let $\mathcal{T}$ denote the triangular objects in the configuration. Consider configurations with $y_1\geq 0$ and minimize the area of their convex hull. Minimal-configuration conjecture. The optimal configuration of $\mathcal{L}$, $\mathcal{T}$ and $\mathcal{T}$ must have the following properties: (1) the right-most vertex of $\mathcal{T}$ coincides with the point $(1,0)$ of $\mathcal{L}$; (2) the top-most vertices of $\mathcal{S}$ and $\mathcal{T}$ coincide. The properties are suggested by numerical grid-search experiments in the study of lower bounds for Moser's worm problem; the supplied text gives no proof or resolution, so the conjecture remains open.