Relative integral identity for equivariant functions on a vector bundle

Let $f:\mathbf{V}(\mathscr E)\rightarrow\mathbf{A}^1$ be a $\mathbf{G}_m$-invariant morphism. Let $r$ be the rank of the vector bundle $\mathbf{V}(\mathscr E^+)$ over $\mathbf{V}(\mathscr E^0)$, and let $\pi^+$, $\sigma$, $f^0$, and $\psi_f$ have the meanings from the preceding setup. The motivic nearby-cycle integral is taken over the indicated vector-bundle projection.

Relative integral identity. The equality

$$\int_{\pi^+}(\psi_f)|_{\mathbf{V}(\mathscr E^+)_\sigma}=\mathbf{L}^{r}\psi_{f^0}$$

holds in the ring $\mathscr M^{\hat{\mu}}_{\mathbf{V}(\mathscr E^0)_\sigma}$.

This is a relative form of the integral identity originally conjectured by Kontsevich and Soibelman. The supplied text gives no resolution of this relative statement; its status is therefore open.