Topping's conjecture for closed immersed surfaces

Let $\Sigma\subset\mathbb{R}^3$ be an immersed connected closed surface. Topping's conjecture.

$$\frac{1}{\operatorname{diam}(\Sigma)}\int_\Sigma |H| > \pi.$$

This conjecture asks for the optimal lower bound for the total absolute mean curvature in terms of the extrinsic diameter; the paper discusses its relation to geometric inequalities and proves it in the axisymmetric setting, while the general case remains open.