ds.dynamical systems – Does an “almost weak mixing” transformation admit a non null ergodic component?

Problem set up:

Let $mathbf X := (X, mathcal A, mu)$ be a standard probability space.

We say that a measure preserving transformation $T$ on $mathbf X$ is $varepsilon$-almost weak mixing if for every $delta > varepsilon$, and every pair of non-null measurable sets $A, B in mathcal A$, there exists an $N > 0$ such that for all $n > N$, we have $frac{1}{n} sum_{k = 1}^{n} |mu(T^{-k}A cap B) – mu(A)mu(B)| < delta mu(A)mu(B)$.

We say a measure preserving transformation $G$ on $X$ admits an ergodic component if there exists some non-null measurable subset $E$ of $X$ such that $G(E) subset E$, and the “restricted system” ($mathbf E, G_{|E})$, with $mathbf E := (E, mathcal A_{|E}, mu_{|E})$ is ergodic. Here $mathcal A_{|E}$ is the restricted sigma algebra, and $mu_{|E}$ is defined by $mu_{|E}(A) := mu(A cap E)/mu(E)$.

Question: Does there exist some $varepsilon > 0$ such that any $varepsilon$-almost weak mixing transformation $T$ on $mathbf X$ admits an ergodic component?

Remark: This is a potential sharpening of an earlier result.