In a volume entitled "Real Analytic and Algebraic Geometry" (ed. F. Broglia, M. Galbiati, and A. Tognoli), there is a paper by Coste and Shiota which proves that if the data in Thom's first isotopy lemma are semi-algebraic, then there exists a semi-algebraic trivialization (in the classical proofs involving integration of controlled vector fields, the flows will not in general be semi-algebraic even if the vector fields are, so this Coste-Shiota result is rather refined).

I'm wondering whether anyone has proved an o-minimal generalization of this result: that if the data of the first isotopy lemma are definable, then there will be a definable trivialization? Guides to relevant literature are highly appreciated!