What is the maximum number of edges of an $n$-vertex [linklessly embeddable](https://en.wikipedia.org/wiki/Linkless_embedding) graph?

A more general question is the following. What is the maximum number of edges of an $n$-vertex graph with [Colin de Verdière number](https://en.wikipedia.org/wiki/Colin_de_Verdi%C3%A8re_graph_invariant) $\mu$?

A related question would be the following. Is there a fixed space (say a 2-manifold) such that graphs which embed linklessly into $\mathbb{R}^3$ are characterized by embedding into that space? This is purely out of curiosity.

Thanks.