graph theory – Can Tutte embedding be guaranteed that each face is convex?

In graph drawing and geometric graph theory, a Tutte embedding of a simple 3-vertex-connected planar graph is a crossing-free straight-line embedding with the properties that the outer face is a convex polygon and that each interior vertex is at the average (or barycenter) of its neighbors’ positions. Tutte embedding

But I don’t know if this means that such embedding when first we fix any
arbitrary outer face (may be convex)
can guarantee that each face is convex.

For example, will the following non-convex face $f_1$ appear in Tutte embedding embedding? If it exists, is there a way to make each face convex?

enter image description here

References:

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