This question surprises me: Is there a non-Platonic convex equilateral dodecahedron (with 1-skeleton the dodecahedral graph)?
The endo-dodecahedron is a non convex example (photo from Wolfram).
The cube is an example of an equilateral polyhedron which has equilateral deformations.
https://mathworld.wolfram.com/EquilateralPolyhedron.html
https://mathoverflow.net/questions/434771/is-the-dodecahedron-flexible-as-a-polytope-with-fixed-edge-lengths
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