Central limit conjecture for quantum Markov-chain fluctuations

Let $X\in\mathcal{A}\otimes\mathcal{B}$ be a selfadjoint operator with $\langle X\rangle_{ss}^{out}=0$. Let $\mathbb{F}_n(X)$ denote the associated fluctuations operator, and let $(X,X)_V$ be its Markov covariance. Central limit conjecture. The fluctuations operator satisfies the central limit theorem

$$\mathbb{F}_n(X)\overset{\mathcal{D}}{\longrightarrow}N(0,V_X),$$

where $N(0,V_X)$ is the centred Gaussian distribution with variance $V_X=(X,X)_V$, and the convergence holds as $n\to\infty$ in distribution with respect to the state $|\varphi\otimes\chi^{\otimes n}\rangle$. This conjecture would establish a quantum central limit theorem for fluctuation operators; the paper states that it is not investigated there and identifies it as an open problem.