Is there a simple proof for the 4-color theorem when restricted to (finite) maps all of whose regions are pentagons? I am in fact most interested in convex pentagons, if that additional structure helps. These maps lead to graphs of maximum degree $\Delta=5$, so a result for such graphs will answer my question.

If all regions are quadrilaterals, then four colors are sometimes necessary,
even if all quadrilaterals are convex, e.g.,

But here there is a simple proof of 4-colorability:
Identify a quadrilateral with an exposed edge, remove it, 4-color the remainder
by induction, and replace the quad, coloring it with a color different from its
at most three neighbors.
Obviously this simple proof fails for pentagons.

Especially the $\Delta=5$ case is likely known to the experts. So a reference would suffice. Thanks!