The $C^k$-map versus ring-homomorphism conjecture

Let $X$ and $Y$ be $C^k$-manifolds and let $E$ be a complex $C^k$ vector bundle of rank $r$ on $X$. Given a correspondence

$$\varphi^{\sharp}:C^k(Y)\longrightarrow C^k(\operatorname{End}_{\Bbb C}(E)),$$

The $C^k$-map versus ring-homomorphism conjecture. The following statements are equivalent: (1) $\varphi^{\sharp}$ is a ring-homomorphism over ${\Bbb R}\hookrightarrow{\Bbb C}$; (2) $\varphi^{\sharp}$ is a weakly $C^k$-admissible ring-homomorphism over ${\Bbb R}\hookrightarrow{\Bbb C}$; and (3) $\varphi^{\sharp}$ is a $C^k$-admissible ring-homomorphism over ${\Bbb R}\hookrightarrow{\Bbb C}$. Thus, any of these equivalent types of correspondence defines the same notion of a $C^k$-map $(X^{\!A\!z},E)\to Y$.