Is Gauss-Lucas' theorem also true for real quaternions?

Question

In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. In complex analysis, the Gauss–Lucas theorem gives a geometric relation between the roots of a polynomial P and the roots of its derivative P'. The set of roots of a real or complex polynomial is a set of points in the complex plane. The theorem states that the roots of P' all lie within the convex hull of the roots of P, that is the smallest convex polygon containing the roots of P. When P has a single root then this convex hull is a single point and when the roots lie on a line then the convex hull is a segment of this line. Now, assume that $\mathbb H$ be the ring of quaternions as a subspace of real four-dimensional space. Consider that $P \in \mathbb H[x]$ is a polynomial. Let $A\subseteq \mathbb H $ be the set of roots of $P$. Is it true that the roots of $P'$ all lie within the convex hull of the roots of $P$? Note that A is not necessarily finite. When $P \in \mathbb R[x]\subseteq \mathbb H[x]$ and $A$ be the set of roots of $P$ in $ \mathbb H $, It is possible to prove the truth of this claim.