Number operator representation conjecture for quantized time evolution

Let $f:\mathbb{C}^{m}\to\mathbb{R}$ be a classical Hamiltonian, let $\mathcal{E}_{b}(f(\vec{Z}))$ be its quantization, and define

$$U_{f,t}:=\exp[-\mathrm{i}t\mathcal{E}_{b}(f(\vec{Z}))],\qquad t\in\mathbb{R}.$$

Let $N_b$ be the number operator and $E_b$ the projection associated with the quantization procedure. Number operator representation conjecture. If $f$ is physically reasonable, then $\mathcal{E}_{b}(f(\vec{Z}))$ is essentially self-adjoint and, for every $t\in\mathbb{R}$ and $\varphi\in\operatorname{ran}E_b$,

$$\lim_{\nu\to\infty}\exp[-\mathrm{i}tf(\vec{Z})-\nu N_b]\varphi=U_{f,t}\varphi.$$

Equivalently,

$$\mathop{\mathrm{s-lim}}_{\nu\to\infty}\exp[-\mathrm{i}tf(\vec{Z})-\nu N_b]=U_{f,t}E_b, \qquad t\in\mathbb{R}.$$

This conjecture seeks to represent the quantum evolution generated by the projected quantized Hamiltonian through a strong limit involving the more familiar number operator, without explicitly inserting the projection $\mathcal{E}_b$. The source does not specify what counts as physically reasonable or provide evidence resolving the conjecture.