Classification of conic divisorial ideals of toric rings associated to posets

Let $P$ be a poset, let $\widehat{P}$ be its augmented poset, let $\mathcal{Q}(\widehat{P})$ denote the set of maximal chains of $\widehat{P}$, and let $\mathfrak{C}_P$ and $\mathfrak{X}_P$ be the polyhedral sets defined from the circuits and general X-shape subposets of $\widehat{P}$, respectively. Write $\mathcal{C}_P$ for the associated toric configuration and $\Bbbk[\mathcal{C}_P]$ for its toric ring. Classification conjecture. The conic divisorial ideals of $\Bbbk[\mathcal{C}_P]$ one-to-one correspond to the points in

$$\mathfrak{C}_P\cap \mathfrak{X}_P\cap \mathbb{Z}^{\mathcal{Q}(\widehat{P})\setminus \{Q_0\}}.$$

This claim gives a polyhedral and integral parametrization of the conic divisorial ideals of the toric ring associated with $P$; the supplied text does not indicate whether the statement is proved or remains open.