Applying basic cohomological calculus it is possible to derive that $\mathbb{HP}^2$ cannot be embedded in a Euclidean space of dimension 11 or less.
From other side, applying spinorial cohomological calculus (Mayer integrality theorem), I am deriving that $\mathbb{HP}^2$ cannot be embedded in a Euclidean space of dimension 12.
Also it is well known that $\mathbb{HP}^2$ can be immersed into $\mathbb{R}^{15}$.
Then my question is:
What is the lower $n$ such that $\mathbb{HP}^2$ can be embedded in $\mathbb{R}^n$?