This is a follow-up to the question

Rigidity/flexibility of Nil-, Sol-, $\widetilde{\rm SL}_2$- structures on closed 3-manifolds

From the answers/comments there and from an excellent survey by Bonahon ("Geometric Structures on 3-Manifolds") I now figured out what are the deformation spaces in the cases of Seifert geometries. What still remains unclear to me is the case of Sol-geometry.

So, let $M$ be a closed 3-manifold admitting Sol-geometry. Thereby, it is a torus bundle over circle with Anosov monodromy. We say that two Sol-structures on $M$ are equivalent if they differ by a diffeomorphism isotopic to identity. Can there be non-equivalent Sol-structures on $M$? If yes, what is known about the deformation space?

(Note that the respective section of Bonahon's survey about rigidity/flexibility is called "Seifert geometries and Sol", however, unfortunately, it seems to me that despite the title, the case of Sol-geometry is omitted there.)