I would be grateful if you could help me with the following question:

Let $n\in \mathbb{N}$ and $k\in \{1, \ldots, n+1\}$. For every $i=1, ..., k$ let $f_i : \partial_i \Delta^n \rightarrow \partial_i \Delta^n$ be an orientation preserving diffeomorphism of the i-th face of the standard $n$-simplex $\Delta^n$. Suppose $f_i = f_j$ on $\partial_i \Delta^n \cap \partial_j \Delta^n$ for all $i,j$. Does there exist a diffeomorphism $f: \Delta^n \rightarrow \Delta^n$ restricting to $f_i$ on $\partial_i \Delta^n$ for every $i$ ? Simplices are considered as manifolds with corners.

I think I can solve it for $k=n+1$ when $f_i$ are time one flows of some vector fields by reducing to the problem of extending a smooth function from a closed set. However, I am interested in the case when $f_i$ are arbitrary diffeomorphisms and $k=n$ so that there should not be any topological obstructions for the extension. I do not know how to start proving it except for trying to write down an explicit formula for the extension.

Thank you very much for your comments.