Fourier-transform conjecture for equivariant quantum volumes and reduced quantum volumes

Let $X$ be a Hamiltonian $T$-manifold with moment map $mu$, and let $X//_tT=mu^{-1}(t)/T$ be the smooth symplectic reduction. Let $Pi_X^{\rm eq}$ denote the equivariant quantum volume of $X$, and let $Pi_{X//_tT}$ denote the quantum volume of the reduction. Define the Duistermaat–Heckman form by

$$\widehat{\omega}=\omega-\lambda\cdot\mu,$$

where $\lambda\in\operatorname{Lie}(T)$, and let $\omega_{\rm red}$ be the reduced symplectic form. Fourier-transform conjecture. The equivariant quantum volume $Pi_X^{\rm eq}(-[\widehat{\omega}])$, viewed as a function of $\lambda\in\operatorname{Lie}(T)$, and the quantum volume $Pi_{X//_tT}(-[\omega_{\rm red}])$, viewed as a function of $t\in\operatorname{Lie}^*(T)$, are related by Fourier transformation:

$$\Pi_X^{\rm eq}(-[\widehat{\omega}])\quad\underset{\rm FT}{\longleftrightarrow}\quad\Pi_{X/\!/_{t}T}(-[\omega_{\rm red}]).$$

This is proposed as a naive conjectural form of Fourier analysis relating equivariant quantum data to the quantum geometry of symplectic reductions; the source gives no resolution status.