Matroid-rooted bounded regular packing conjecture for mixed hyperarborescences

Let $\mathcal{F}=(V,\mathcal{E}\cup\mathcal{A})$ be a mixed hypergraph, let $k\in\mathbb{Z}_{+}$, let $f,g\in\mathbb{Z}_{+}^{V}$, let $S$ be a multiset of vertices in $V$, and let $M=(S,r_M)$ be a matroid. For $U,W\subseteq V$, write $S_U$ for the elements of $S$ rooted in $U$, and let $g_k$ denote the function used in the bounded-packing conditions. A subpartition $\mathcal{P}$ of $W$ is a family of pairwise disjoint nonempty subsets of $W$.

Matroid-rooted bounded regular packing conjecture. There exists an $M$-rooted $(f,g)$-bounded $k$-regular packing of mixed hyperarborescences in $\mathcal{F}$ if and only if the conditions and hold and, for all $U,W\subseteq V$ and every subpartition $\mathcal{P}$ of $W$,

$$e_{\mathcal{E}\cup\mathcal{A}}(\mathcal{P})+r_M(S_U)+g_k(W-U)\geq k|\mathcal{P}|+f(U-W).$$

This conjecture seeks the mixed-hypergraph analogue of the established characterization for mixed graphs. The source identifies it as an open problem; it would follow from the proposed orientation extension in the second conjecture.