Resolution-of-unity conjecture for coherent states on Lie groups

Let $G$ be a Lie group, let $\mathfrak g$ be its Lie algebra, let $\Phi(g,X)$ denote the phase-space label associated with $(g,X)\in G\times\mathfrak g$, and let $\Psi^t_{\Phi(g,X)}$ be the corresponding coherent state. For sufficiently small $t>0$, there should exist a constant $C_t$ such that

$$\mathds{1}=C_t\int_G\int_{\mathfrak g}|\Psi^t_{\Phi(g,X)}\rangle\langle\Psi^t_{\Phi(g,X)}|\,\bigl(\langle\Psi^t_{\Phi(g,X)}|\Psi^t_{\Phi(g,X)}\rangle\bigr)^{-1}\,dg\,dX,$$

with

$$C_t^{-1}=\operatorname{vol}(G)\int_{\mathfrak g}\bigl(\langle\Psi^t_{\Phi(g,X)}|\Psi^t_{\Phi(g,X)}\rangle\bigr)^{-1}\,dX\propto t^{-n}.$$

Resolution-of-unity conjecture. For sufficiently small $t>0$, the displayed resolution of unity and the stated asymptotic scaling of $C_t$ hold for an arbitrary Lie group $G$. If $G$ is commutative or $G=SU(2)$, they hold for every $t>0$. The claim proposes a normalized coherent-state resolution using the inverse coherent-state norm rather than the heat-kernel measure. The paper identifies the commutative case and $SU(2)$ as cases where the assertion is expected to hold for all positive $t$, while the general small-$t$ assertion remains unproved in the supplied text.