Toric nef-value fibration conjecture beyond the smooth Gorenstein case

Let $(X,\mathcal L)$ be an $n$-dimensional toric polarized variety, not necessarily smooth or Gorenstein, and let $\mu(\mathcal L)$ and $\tau(\mathcal L)$ denote the corresponding effective log threshold and nef value. Toric nef-value fibration conjecture. If

$$\mu(\mathcal L)>\frac{n+1}{2},$$

then

$$\mu(\mathcal L)=\tau(\mathcal L).$$

The conjecture extends the smooth-variety threshold criterion to arbitrary toric polarized varieties; in the non-Gorenstein case the source defines these invariants through the associated polytope. No resolution is given in the supplied text.