# Rank-respecting augmentation of hypergraphs with negamodular constraints

Given a crossing negamodular function $R:2^V\to \mathbb{Z}$ such that $R(X)\ne 1$ for every $X\subseteq V$ and a hypergraph $G_0=(V,\mathcal{E}_0)$, find a hypergraph $G=(V,\mathcal{E})$ of minimum total size such that $d_G(X)\ge R(X)-d_{G_0}(X)$ holds for every $X\subseteq V$ and the rank of $G$ does not exceed the rank of $G_0$.
This problem is open for graphs (i.e. when $G_0$ is a graph).