It is well known that $\mathbb{CP}^4$ cannot be embedded in a Euclidean space of dimension 11 or less.
Similarly, at question was proved that:
$\mathbb{CP}^4$ does not immerse into $\mathbb{R}^{12}$.
For other side at question was proved that:
$\mathbb{CP}^4$ does not immerse into $\mathbb{R}^{13}$.
My question now is: What is the lower $n$ such that $\mathbb{CP}^4$ can be embedded in $\mathbb{R}^n$?