Exponential stabilization conjecture for coupled spin-$J$ systems with state feedback

Let $J$ be the spin parameter, let $N=2J+1$, and let $(\rho_t,\hat{\rho}_t)$ denote the coupled spin-$J$ system and its estimated state, with initial condition $(\rho_0,\hat{\rho}_0)\in\mathcal S_N\times\mathcal S_N\setminus\{(\boldsymbol\rho_{\bar n},\boldsymbol\rho_{\bar n})\}$ and measurement efficiency $\eta\in(0,1)$. For $\alpha>0$ and $\beta\geq1$, consider the feedback controllers

$$u_{\bar n}(\hat{\rho})=\alpha\bigl(1-\operatorname{Tr}(\hat{\rho}\boldsymbol\rho_{\bar n})\bigr)^\beta$$

and

$$u_{\bar n}(\hat{\rho})=\alpha\bigl(J-\bar n-\operatorname{Tr}(J_z\hat{\rho})\bigr)^\beta.$$

Exponential stabilization conjecture. The first controller almost surely exponentially stabilizes $(\rho_t,\hat{\rho}_t)$ to $(\boldsymbol\rho_{\bar n},\boldsymbol\rho_{\bar n})$ when $\bar n\in\{0,2J\}$, with sample Lyapunov exponent at most $-\eta M$. The second controller almost surely exponentially stabilizes the same target for every $\bar n\in\{0,\dots,2J\}$, with sample Lyapunov exponent at most $-\eta M/2$ for $\bar n\in\{1,\dots,2J-1\}$ and at most $-\eta M$ for $\bar n\in\{0,2J\}$. This proposes exponential almost-sure stabilization when the initial state and estimate are not already at the target; the source motivates it by results proved when $\rho_0=\hat\rho_0$, while the asserted extension to distinct initial states remains unverified.