Cattani–Cox–Dickenstein's Codimension One conjecture for toric varieties

Let $X$ be a complete simplicial toric variety with Cox ring $B(\Sigma)$, and let $F_0,\ldots,F_n\in B(\Sigma)$ be homogeneous polynomials that do not vanish simultaneously on $X$. Let $\beta_0$ be the anticanonical class of $X$, and define the critical degree

$$\omega=\sum_i\deg F_i-\beta_0.$$

Cattani–Cox–Dickenstein's Codimension One conjecture. The critical-degree component of the quotient by the $F_i$ should satisfy

$$\dim_{\mathbb C}\left(S_\omega/(F_0,\ldots,F_n)_\omega\right)=1.$$

This conjecture concerns the top-degree piece of a quotient in the Cox-ring setting for complete simplicial toric varieties. The supplied text does not indicate whether the conjecture has been resolved.