Hauswirth–Pérez–Romon–Ros isoperimetric conjecture for flat torus products

Let $\mathbb{T}^2$ be a flat $2$-torus, and consider the isoperimetric problem in $\mathbb{T}^2\times\mathbb{R}$.

Hauswirth–Pérez–Romon–Ros conjecture. The only solutions of the isoperimetric problem in $\mathbb{T}^2\times\mathbb{R}$ are spheres, cylinders, and pairs of horizontal planes.

The conjecture concerns the classification of isoperimetric minimizers in a flat torus product. The paper constructs a counterexample, showing that a Lawson-type family can have smaller area than the conjectured competitors near the transition from cylinders to pairs of planes.