Consider an abstract stratified set $(V, \Sigma)$ in the sense of Thom-Mather (see Mather's note page 491-492 https://www.ams.org/journals/bull/2012-49-04/S0273-0979-2012-01383-6/S0273-0979-2012-01383-6.pdf).
The first isotopy lemma, proved by considering controlled vector fields which admits a continuous local flow, making the local trivializations into a continuous map. As a corollary we get that the tubular neighborhood $T_X$ of a strata $X \in \Sigma$ is stratified-homeomorphic, over a small open subset $U \subset X$, to a bundle of cone, namely $$ T_X \simeq U \times c(L_X,\pi_X) $$ where $c(L_X,\pi_X)$ is the cone $L_X \times [0, +\infty)/ \sim$ where $(a,t) \sim (b,s)$ if $\pi_X(a) = \pi_X(b)$ and $t=s=0$ and $L_X = \rho_X^{-1}(1)$ where $(T_X, \pi_X, \rho_X)$ are control data for the stratum $X$.
The question is: can we precise a better regularity in this general context ?
More precisely, over a chosen stratum the flow of a controlled vector field is smooth and non-continuity arises when we pass from a stratum to an other. Is this "isomorphism of stratified set $h$" in the proof of the key lemma 10.2 of the above link can be chosen to be an homeomorphism that restrict to a diffeomorphism over any strata ?
Note: in the link the definition of isomorphism is a bit unclear to me (p.492), we could for instance have two isomorphic abstract stratifications with homeomorphic strata but non-diffeomorphic strata, is there a reason to allow this ? (if I remember some exotic sphere appears as boundary of the Milnor fiber of isolated singularity of complex hypersurface, which seems to be the link in our vocabulary).
