Weak mixing sequence conjecture for functions satisfying the correlation condition

Let $(X,\mathscr{B},\mu,T)$ be a measure-preserving system, let $U$ denote its Koopman operator, and let $f\in L^2(X,\mu)$ satisfy

$$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^N\left|\langle U^n f,g\rangle\right|=0$$

for every $g\in L^2(X,\mu)$. A sequence $(z_n)_{n=1}^{\infty}$ is weakly mixing if its correlations have the corresponding Cesàro convergence to zero. The weak mixing sequence conjecture. Under these assumptions, for almost every $x\in X$, the sequence $(f(T^n x))_{n=1}^{\infty}$ is weakly mixing. The conjecture asks whether the stated Hilbert-space correlation condition on $f$ suffices for almost-everywhere weak mixing of its pointwise orbit sequence; the surrounding discussion explains that this would remove the need for the underlying system to be weakly mixing in the relevant pointwise ergodic theorem.