Octonion-valued multilinear polynomial image conjecture

Let $p(x_1,\dots,x_n)$ be a non-associative, non-commutative multilinear polynomial over $\mathbb{R}$. Evaluate $p(x_1,\dots,x_n)$ on the real octonion algebra $\mathbb{O}$.

Octonion image conjecture. The image of this evaluation is a real vector subspace of $\mathbb{O}$.

The source motivates this question by the role of the octonions as an $8$-dimensional real division algebra and by known analogous results for the quaternions. It describes the octonion assertion as a conjecture for a paper being prepared; no resolution is given.