Finite-horizon approximation conjecture for extremum-seeking systems

Let $x(t)$ be a trajectory of systems and, and let $\bar{x}(t)$ be a trajectory of the averaged approximation

$$\dot{\bar{x}} = f(\bar{x},t) + \epsilon g_{2n_e}(\bar{x},t)\left ( \alpha \omega \right )^{\frac{2n_e}{2m+1}}B_{n_e} - k\alpha^{\frac{2n_o+1}{2m+1}}\omega^{\left ( \frac{2n_o+1}{2m+1}-1 \right )} A_{n_0} \frac{g_{n_o}(\bar{x},t)g^T_{n_o}(\bar{x},t)}{2^{4n_o+1}} \left (\frac{\partial V}{\partial \bar{x}}\right )^T,$$

where

$$A_{n_0} = \sum_{l=0}^{n_o}\binom{2n_o+1}{l}^2, \qquad B_{n_e} = \frac{1}{2^{2n_e}}\binom{2n_e}{n_e}.$$

Finite-horizon approximation conjecture. For any $\delta>0$, any compact set $K \subset \mathbb{R}^n$, and any $t_0,T \in \mathbb{R}_{\geq 0}$, there exists $\epsilon^\star(\delta,K,T)>0$ such that, for all $|\epsilon|<\epsilon^\star$, there exists $\omega^\star$ such that for each $\omega>\omega^\star$,

$$\max_{t\in [t_0,t_0+T]}\|x(t)-\bar{x}(t)\|<\delta.$$

This conjecture asserts uniform approximation of the original and averaged trajectories on arbitrary finite time intervals under sufficiently small perturbation amplitude and sufficiently high excitation frequency. The supplied text does not state whether the claim has been proved or remains open.