The maximal-set realization conjecture for Borel-fixed ideals

Let $\textnormal{\textsf{S}}_{p(t)}^{n}$ be the set of Borel-fixed ideals with Hilbert polynomial $p(t)$, and let $\mathcal{M}\mathcal{C}(J_1,\ldots,J_s)$ denote the common maximality cone associated with ideals $J_1,\ldots,J_s$. Let $\succeq_\Omega$ be the relation induced by a term order $\Omega$.

Maximal-set realization conjecture. For every set of ideals $J_1,\ldots,J_s\in\textnormal{\textsf{S}}_{p(t)}^{n}$ such that

$$\mathcal{M}\mathcal{C}(J_1,\ldots,J_2)\neq\emptyset,$$

there exists a term order $\Omega$ such that

$$\{J_1,\ldots,J_s\}=\max_{\succeq\hspace{-2.5pt}\succeq_\Omega}\textnormal{\textsf{S}}_{p(t)}^{n}.$$

This is presented as a restatement of the preceding conjecture, expressing the claim directly in terms of sets of ideals whose common maximality cone is nonempty. The source gives no evidence that it has been resolved.