Isomorphism between the normal-space quotient and the tangent bundle of an orbit space

Let $M$ be a smooth manifold with a proper action of a group $G$. For each $x\in M$, let $N_x(M,G\cdot x)$ be the normal space to the orbit through $x$, and define

$$A:=\bigcup_{x\in M}N_x(M,G\cdot x).$$

Let $G_x$ be the stabilizer of $x$, set

$$\widetilde{A}:=\bigcup_{x\in M}A_x^{G_x},$$

and endow $\widetilde{A}/G$ with its induced stratified vector-bundle structure over $M/G$. Let $T(M/G)$ denote the stratified tangent bundle of the Whitney A stratified space $M/G$. Normal-space quotient isomorphism conjecture. The canonical morphism of stratified vector bundles

$$\widetilde{A}/G\to T(M/G)$$

is an isomorphism over $M/G$. The map is bijective and restricts to an isomorphism of smooth vector bundles on each stratum; the conjectural content is that it is an isomorphism of stratified vector bundles globally.