Araujo stated the following four open questions at the end of his paper, page $518$ and $519.$

Question $1:$ Assume that there exists a biseparating map $T:A^n(\Omega:E)\to A^m(\Omega',F)$ which is not linear. Can we deduce that the support map $h:\Omega'\to\Omega$ is a diffeomorphism of class $n?$

Problem $2:$ Suppose that $C^\infty(\Omega,E)$ is the space of $E$-valued functions which are of class $C^\infty$ in $\Omega,$ and that $C^\infty(\Omega',F)$ is defined in a similar way. Describe the linear biseparating maps from $C^\infty(\Omega,E)$ onto $C^\infty(\Omega',F).$ Must such map be continuous?

Problem $3:$ Let $\Omega$ and $\Omega'$ be unbounded open subsets of $\mathbb{R}^p$ and $\mathbb{R}^q$ respectively. Describe the linear biseparating maps from $C^n_*(\Omega,E)$ onto $C^m_*(\Omega',F).$

Problem $4:$ Determine all subspaces $A(\Omega,E)\subseteq A^n(\Omega,E)$ and $B(\Omega',F)\subseteq A^m(\Omega',F)$ such that the existence of a linear biseparating map from $A(\Omega,E)$ onto $B(\Omega',F)$ implies that $E$ and $F$ are isomorphic as Banach spaces.

What is the status of these questions? Any progress on each of them? If yes, any reference is appreciated.

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Definitions: Let $\Omega\subseteq \mathbb{R}^p$ be an open subset and $E$ be a Banach space. Denote $C^n(\Omega,E)$ the space of $E$-valued functions $f$ on $\Omega$ that are of class $C^n.$ $T:C^n(\Omega,E)\to C^m(\Omega',F)$ is called biseparating map if $T$ is bijective and $$\|Tf\| \|Tg\|=0 \text{ if and only if }\|f\|\|\|g\|=0$$