Existence of imaginary quadratic fields of class numbers coprime to $p$ with prescribed splitting behaviour of $p$

Let $x\in\{\text{totally ramified, inert, totally split}\}.$

If $p\geq 5$ is a prime, are there infinitely many imaginary quadratic fields $K=\mathbb{Q}(\sqrt{-d})$ of class number coprime to $p$ so that $p$ has ramification behaviour $x$ in $K/\mathbb{Q}$?