Minimal rank of primitively $n$-universal integral quadratic forms over local rings

Let $F$ be a local field and let $R$ be its ring of integers. For a positive integer $n$, an integral quadratic form defined over $R$ is called primitively $n$-universal if it primitively represents all quadratic forms of rank $n$. It was proved in arXiv:2005.11268 that the minimal rank of primitively $1$-universal quadratic forms over the $p$-adic integer ring $\mathbb{Z}_p$ is $2$ if $p$ is odd, and $3$ otherwise. In this article, we completely determine the minimal rank of primitively $n$-universal quadratic forms over $R$ for any positive integer $n$ and any local ring $R$ such that $2$ is a unit or a prime.