With the usual topology on $\Bbb R$, a compactification $\mathrm{id}_{\Bbb R}:\Bbb R\to v\Bbb R$ can have a remainder $v\Bbb R \setminus \Bbb R$ of cardinality $1,2, 2^{\aleph_0}=c,$ or $2^c.$ The only possibilities less than $c$ are $1,2.$

Suppose $c^+<2^c.$ What possible cardinals between $c$ and $2^c$ can be the cardinals of such remainders?

Is there perhaps a Forcing argument that can answer or partly answer this?