BZSV's Plancherel algebra conjecture for hyperspherical Hamiltonian spaces

Let $G$ be a group and $X$ a $G$-space such that the Hamiltonian space $T^*X$ is hyperspherical. Let $\PL_{X,\hbar}$ be the non-commutative Plancherel algebra over $k[\hbar]$, and let $\PL_X$ be the Plancherel algebra without loop rotation. Let $\Mc$ be the hyperspherical dual of $T^*X$, and let $\Mc^{\shear}$ denote its shearing.

BZSV's Plancherel algebra conjecture. The following properties hold: $\PL_{X,\hbar}$ is flat over $k[\hbar]$; $\PL_X$ is commutative; and there is an isomorphism of $\Gc$-equivariant commutative 2-shifted Poisson algebras

$$\PL_X\isom \cO(\Mc^{\shear}).$$

Moreover, there is a canonical isomorphism

$$\PL_{X,\hbar}\isom \cO_{\hbar}(\Mc^{\shear})$$

between the quantizations of the two Poisson algebras.

This conjecture identifies the Plancherel algebra with functions on the hyperspherical dual and predicts that the loop-rotation version provides its quantization. The source attributes the conjecture to BZSV; its resolution status is not specified here.