Suppose a probability perserving system $(X,\mathcal{B}_X,\mu,T)$ is weakly mixing (here $\mathcal{B}_X$ is Borel algebra of topology space $X$ and $\mu$ is a finite/probability measure), i.e. $$ \frac{1}{N}\sum_{n=0}^{n=N-1}|\mu(A\cap T^{-n}B) - \mu(A)\mu(B)| \xrightarrow{N\to+\infty} 0 \quad \forall A,B \in \mathcal{B}_X $$ And there are many equivalent description of weakly mixing. In Furstenberg's ergodic proof of Szemerédi's theorem (One can refer Furstenberg's paper in 1977) he proved weakly mixing implies "weakly mixing on all order for sets" like $$ \frac{1}{N}\sum_{n=0}^{n=N-1}|\mu(T^{-n}A_1 \cap T^{-2n}A_2 \cap \cdots \cap T^{-kn}A_k) - \mu(A_1)\cdots \mu(A_k)| \xrightarrow{N\to+\infty} 0 \quad \forall A_1,..,A_k \in \mathcal{B}_X\tag*{(A)} $$ In almost all existing references on this topic a so-called "van der Corpt trick " on Hilbert space is used and by that technique one can prove stronger results than (A) for commutative weakly mixing operators. etc.
I suspect there is a more-straight way to prove (A) because using induction it is sufficient to show weakly mixing implies $$ \frac{1}{N}\sum_{n=0}^{n=N-1}|\mu(T^{-n}A_1 \cap T^{-in}A_2 \cap T^{-jn}A_3) - \mu(A_1)\mu(A_2) \mu(A_3)| \xrightarrow{N\to+\infty} 0 \quad \forall A_1,A_2,A_3 \in \mathcal{B}_X,i,j\in\mathbb{N}^+\tag*{(B)} $$ which means something like "3-weakly mixing".
If one can prove (B) then using some estimations on measures one can prove (A) by induction.
My question is does there exist a direct way to get (B)?