The generalised Casson volume inequality conjecture

Let $M$ be a cusped finite-volume hyperbolic three-manifold, let $\mathbf{s}$ be a collection of signed slopes, and let $\mathcal{T}$ be an ideal triangulation of $M$. Let $\mathcal{A}(\mathcal{T},\mathbf{s})$ be the open polytope of angle structures: assignments of angles in $(0,\pi)$ to the model edges satisfying the tetrahedron, edge, and signed-slope angle equations. Let $\mathcal{V}:\mathcal{A}(\mathcal{T},\mathbf{s})\to\mathbb{R}$ be the sum of the Lobachevsky volumes of the tetrahedra, and let $M(\mathbf{s})$ denote the Dehn filling along the signed slopes.

The generalised Casson conjecture. For every $\alpha\in\mathcal{A}(\mathcal{T},\mathbf{s})$,

$$\mathcal{V}(\alpha)\leq \operatorname{Vol}(M(\mathbf{s})).$$

The inequality would bound the volume functional on every angle-structure polytope by the volume of the corresponding Dehn filling. The paper gives examples showing that this proposed approach to the generalised Casson conjecture cannot succeed, so the conjecture is refuted.