Asymptotic approximation of population covariances by linearized recursions

Let $\Omega$, $\Phi$, and $\Psi$ be the population covariances appearing in Theorem, and let $\Omega_L^\mathrm{lin}$, $\Phi_{\tilde{L}}^\mathrm{lin}$, and $\Psi_{\tilde{L}}^\mathrm{lin}$ be the final iterates of the linear recursions from Definition. The Frobenius-norm errors satisfy

$$\norm{\Omega-\Omega_L^\mathrm{lin}}_F + \norm{\Psi-\Psi_{\tilde{L}}^\mathrm{lin}}_F + \norm{\Phi-\Phi_{\tilde{L}}^\mathrm{lin}}_F \lesssim 1.$$

Asymptotic covariance approximation conjecture. The population covariances can be asymptotically approximated by the last iterates of the corresponding linear recursions, with the displayed combined error bound. The source provides no resolution or further context establishing whether this claim is proved or remains open.