Generic maximal nonnegative rank for hypersimplices

For $n\geq 5$ and $2\leq k\leq n-2$, let $\mathcal{R}_{n,k}$ denote the realization space of the $(n,k)$-hypersimplex, and call a realization combinatorial when it has the hypersimplex's prescribed combinatorial type. The nonnegative rank of such a realization is denoted by $\mathrm{rk}_+$.

Generic maximal-rank conjecture. The combinatorial hypersimplices of nonnegative rank $2n$ form a dense open subset of $\mathcal{R}_{n,k}$.

The claim extends the proved result for $k=2$ to all $2\leq k\leq n-2$. The preceding discussion says that the rank-$2n$ locus is already nonempty and open in the relevant cases, while density for higher values of $k$ remains unproved.