Equality of the Schwartz index and the Euler obstruction

Let $(V,0)\subset(\mathbb{C}^n,0)$ be a complex analytic germ, let $W$ be a local ambient representation, and let $\tilde{\nu}$ be a continuous vector field on $W$ with an isolated singularity at $0$ inducing a stratified vector field $\nu$ on $V$. Write $\mathds{1}_V$ for the indicator function of $V$. Equality of the Schwartz index and the Euler obstruction.

$${\textup{Sch}}(\nu,V,0)=\textup{Eu}(\tilde{\nu},\mathds{1}_V,0).$$

The Schwartz index is defined for stratified vector fields with an isolated singularity, while the Euler obstruction of the constructible function $\mathds{1}_V$ is defined microlocally through the characteristic cycle. The equality is presented as a conjecture because it is expected from the microlocal interpretation, but the source states that it was not seriously proved there.