Averaging conjecture for non-vacuum LRS Bianchi III solutions

Let $(H(t),\mathbf x(t))$ be the solution of the quasi-standard system with arbitrary non-vacuum initial data satisfying $\Omega>0$, the Hamiltonian constraint, and $H>0$. Let $\mathbf z(t)$ solve the corresponding averaged equation

$$\dot{\mathbf z}=H(t)\,\overline{\mathbf f}^{1}(\mathbf z),$$

where $\overline{\mathbf f}^{1}$ is the averaged vector field. Let $\mathbf X(t)$ and $\mathbf Z(t)$ be the two-vectors formed from the $\Sigma_+$ and $\Omega$ components of $\mathbf x(t)$ and $\mathbf z(t)$, respectively. Averaging conjecture. There exists $t_*$ such that

$$\mathbf X(t)-\mathbf Z(t)=\mathcal O\bigl(H(t)\bigr)\qquad\forall\,t>t_*.$$

This conjecture would provide the error estimate needed to justify the averaged description of the future behaviour, since the system has only quasi-standard rather than standard form. The source does not state a resolution.