Orientation conjecture for mixed hypergraphs with supermodular and submodular bounds

Let $\mathcal{F}=(V,\mathcal{E}\cup\mathcal{A})$ be a mixed hypergraph, let $h$ be an integer-valued intersecting supermodular function on $V$, and let $b$ be a submodular function on $V$. For a subpartition $\mathcal{P}$ of $V$, let $e_{\mathcal{E}\cup\mathcal{A}}(\mathcal{P})$ count the mixed-hypergraph edges joining distinct parts, and let $\overrightarrow{\mathcal{E}}$ be an orientation of the hyperedges in $\mathcal{E}$.

Mixed-hypergraph orientation conjecture. There exists an orientation $\overrightarrow{\mathcal{E}}$ of $\mathcal{E}$ such that

$$e_{\overrightarrow{\mathcal{E}}\cup\mathcal{A}}(\mathcal{P})\geq\sum_{X\in\mathcal{P}}h(X)-b(\cup\mathcal{P})$$

for every subpartition $\mathcal{P}$ of $V$ if and only if

$$e_{\mathcal{E}\cup\mathcal{A}}(\mathcal{P})\geq\sum_{X\in\mathcal{P}}h(X)-b(\cup\mathcal{P})$$

for every subpartition $\mathcal{P}$ of $V$. This is proposed as an extension of the corresponding mixed-graph orientation theorem and would imply the matroid-rooted bounded regular packing conjecture by the same proof.