Generalization of the joint-probability result for geometric path integral quantization

Let ${\bf M}$ be a complete Kähler manifold satisfying the condition of Proposition as a Riemannian manifold, and let its Kähler form be the symplectic form. Assume that ${\bf M}$ is prequantizable, with a holomorphic prequantization bundle, and let $H:{\bf M}\to\mathbb{R}$ be a classical-mechanical Hamiltonian satisfying some adequate conditions. Generalization conjecture. Corollary can be generalized for such $H$. The claim concerns extending the preceding joint-probability result to Hamiltonians on this geometric quantization setting, but the source does not specify the adequate conditions or the precise form of the generalization.