Pólya functional bounds for bounded convex planar domains

Let $D\subset\mathbb{R}^2$ be a bounded convex planar domain. Let $T(D)$ denote its torsional rigidity, let $\lambda_1(D)$ be the first Dirichlet eigenvalue of $-\Delta_D$, and define the Pólya functional by

$$F(D)=\frac{\lambda_1(D)T(D)}{|D|}.$$

Pólya functional bounds conjecture. For every bounded convex planar domain $D$,

$$\frac{\pi^2}{24}<F(D)<\frac{\pi^2}{12},$$

and both bounds are sharp. The lower bound is approached by a collapsing sequence of isosceles triangles converging to an interval, while the upper bound is approached by elongating rectangles converging to the infinite strip. This conjecture remains open even for triangles.