The homeomorphism extension for thin points and points in the double locus

Let $x$ be a thin point with $f_*={\rm id}$, or let $x\in X_0^2$. In the latter case, let $B^{T_xN_0}(o_x,\epsilon)$ denote the ball of radius $\epsilon$ centered at $o_x$ in the tangent space $T_xN_0$. The preceding theorem gives a homeomorphism involving the ball $B^{\mathbb R^{m-1}}(0,\epsilon)$.

Homeomorphism extension. The homeomorphism version of Theorem would also hold for any thin point $x$ with $f_*={\rm id}$ and for any $x\in X_0^2$ if the ball $B^{\mathbb R^{m-1}}(0,\epsilon)$ is replaced by $B^{T_xN_0}(o_x,\epsilon)$.

This proposes an extension of the preceding almost-isometry result from the local Euclidean ball to the corresponding tangent-space ball, including thin points with trivial induced map and points in $X_0^2$. The supplied text does not state whether this assertion has been proved or remains open.