Microlocalization conjecture for hyperbolic restriction

Let $X$ be the variety with an attracting-point action and chamber decomposition described above, let $\fa=\operatorname{Lie}\sA$, let $\fq=T_x^*X$, and let $\eta:\fa\to\fq$ be the homomorphism induced by the cotangent-space action. For a chamber $\fC$, write $\eta^*\bmu(\operatorname{IC}(X))_\fC$ for the restriction of the microlocalized perverse sheaf to $\fC$, and let $j_\fC^!\iota_\fC^*\operatorname{IC}(X)$ denote hyperbolic restriction.

Microlocalization conjecture. (a) $\eta^*\bmu(\operatorname{IC}(X))$ is constant in any chamber $\fC$. (b) There is a canonical isomorphism

$$\eta^*\bmu(\operatorname{IC}(X))_\fC\iso j_\fC^!\iota_\fC^*\operatorname{IC}(X).$$

The conjecture proposes that microlocalization is locally constant on the chamber decomposition and recovers the corresponding hyperbolic restriction. The source motivates it by results and examples discussed earlier in the paper; its resolution is not specified in the supplied text.