Universally bad zero-Banach-density sequences conjecture

Let $(n_k)$ be a sequence of integers with zero Banach density, and let $(X,\Sigma,\mu,T)$ be an aperiodic dynamical system. For $f\in L^1(\mu)$, define the averages

$$\frac{1}{N}\sum_{k=1}^{N}f(T^{n_k}x).$$

Zero-Banach-density sequences conjecture. There exists some $f\in L^1(\mu)$ such that these averages do not converge almost everywhere.

This conjecture asserts that every zero-Banach-density sequence is universally $L^1$-bad on every aperiodic dynamical system. The question is motivated by the contrast between known universally bad sequences such as $(k^2)$ and the unresolved search for sequences with gaps tending to infinity whose averages converge almost everywhere for every $L^1$ function.