Generic fluctuation for dissipative ergodic averages

Let $(X,\mathcal{B},\mathbb{P},T)$ be a measure-preserving system, and let $\mu_N^T f=\sum_{n\in\mathbb{Z}}\mu_N(n)f\circ T^n$ be a dissipative sequence of operators. Assume that

$$\mu_N^T f\longrightarrow\int f$$

in $L^1$-norm for every $f\in L^1$. Generic fluctuation conjecture. There is a dense $G_\delta$ set $\mathcal{O}\subset L^1$ such that for every $f\in\mathcal{O}$, the averages $\mu_N^T f(x)$ fluctuate infinitely often around $\int f$ for almost every $x\in X$. The statement predicts generic pointwise fluctuation despite $L^1$-norm convergence; the supplied source does not give a resolution, so it remains open.