The $C^k$-map to Euclidean space conjecture

Let $X$ be a $C^k$-manifold and $E$ a complex $C^k$ vector bundle of rank $r$ on $X$. Let $(y^1,\ldots,y^n)$ be a global coordinate system on $\Bbb R^n$, and let

$$\eta:y^i\longmapsto m_i\in C^k(\operatorname{End}_{\Bbb C}(E)),\qquad i=1,\ldots,n,$$

be an assignment such that the $m_i$ commute pairwise, every eigenvalue of $m_i(p)$ is real for every $p\in X$, and the nilpotency of each $m_i(p)$ is at most $k+1$. The $C^k$-map to Euclidean space conjecture. Then $\eta$ extends to a unique $C^k$-admissible ring-homomorphism

$$\varphi_\eta^{\sharp}:C^k(\Bbb R^n)\longrightarrow C^k(\operatorname{End}_{\Bbb C}(E))$$

over ${\Bbb R}\hookrightarrow{\Bbb C}$ and hence defines a $C^k$-map $(X^{\!A\!z},E)\to\Bbb R^n$.