Semirandom pointwise averaging conjecture

Let $(\mathbbm u_n)_{n\geq 1}$ be a decreasing sequence of probabilities, and let $\delta>0$. Suppose

$$\mathbbm u_n\cdot \frac{n}{(\log\log n)^{1+\delta}}\to\infty$$

and

$$\mathbbm u_n\to 0.$$

Semirandom pointwise averaging conjecture. Then the random sequence $R^\omega=\{r_1<r_2<r_3<\dots\}$ satisfies, in every dynamical system, for bounded functions $F_i$,

$$\lim_{N\to\infty}\frac1N\sum_{n<N}T^nF_1(x)T^{r_n}F_2(x)=\overline F_1\,\overline F_2$$

for almost every $x\in X$, where $\overline F_i$ is the projection of $F_i$ to the $T$-invariant functions.

The supplied context gives a related theorem with the stronger condition involving $n^{1/2}/\log^{3+\delta}n$, and explains that the zero-density assumption is necessary for the stated limit. The conjectural statement's resolution is not supplied here.