Immersed spheres made of regular pentagons

Let $M$ be a closed polyhedral surface homeomorphic to the sphere $S^2$ whose faces are all congruent regular pentagons, and suppose that $M$ is immersed in $\mathbb{R}^3$.

For surfaces made of congruent squares, the corresponding question has a positive answer. The Kepler–Poinsot great dodecahedron is an immersed surface with regular pentagonal faces, but it is not a sphere: its Euler characteristic $V - E + F$ is $-6$. Kenyon's question is also of interest for surfaces embedded in $\mathbb{R}^3$.

Problem. Must $M$ be the boundary of a solid obtained by gluing solid dodecahedra together along common faces?