Conjectural volume bounds for toric Gorenstein singularities

Let an $n$-dimensional toric Gorenstein singularity be associated with a polytope $\Delta_{n-1}$, either reflexive or non-reflexive. Let $Y$ be the associated Sasaki–Einstein base manifold, let $V_{n,\min}$ be its minimized volume, and let $\widetilde{X}$ be the complete resolution of the corresponding compact toric variety. Write $\chi$ for the Euler number of $\widetilde{X}$ and $c_1$ for its first Chern class. Volume-bound conjecture. The minimized volume satisfies

$$\frac{1}{\chi}\leq V_{n,\min}<m_n\int c_1^{n-1}.$$

The left-hand bound is saturated when the Gorenstein singularity is an Abelian orbifold of $\mathbb{C}^n$. In the reflexive cases, the coefficients $m_n$ are positive and satisfy $m_3\sim3^{-3}$, $m_4\sim4^{-4}$, and $m_n>m_{n+1}$. These bounds are intended to relate volume minimization to topological invariants of toric resolutions; the source gives no evidence that the conjecture has been resolved.