Rectangular minimizers among convex quadrilaterals

Let $\mathbb{S}$ be the unit-area square, let $C_\epsilon$ denote the collapsing rectangle introduced in the paper, and let $\sigma_1$ and $\mu_1$ be the first Steklov and Neumann eigenvalues, respectively. For $0<C\leq 4\sigma_1(\mathbb{S})$, consider

$$\inf\{\mu_1(\Omega)|\Omega|\mid \Omega\subset\mathbb{R}^2\ \text{is a convex quadrilateral and}\ \sigma_1(C_\epsilon)P(C_\epsilon)=C\}.$$

Rectangular minimizer conjecture. For every $0<C\leq 4\sigma_1(\mathbb{S})$, the solution of this minimization problem is given by a rectangle.

This conjecture predicts the optimizer within the class of convex quadrilaterals, based on the numerical comparison between random convex quadrilaterals and collapsing rectangles. The supplied status evidence marks the candidate as resolved, but gives no resolving reference.