Frantzikinakis--Kuca conjecture on seminorm control along arithmetic progressions

Let $a\colon\mathbb N\to\mathbb Z$ be strictly increasing. The notions of being good for seminorm control and for degree-$(\ell+1)$ seminorm control along $\ell$-term arithmetic progressions are understood for the system $(X,\mathcal X,\mu,T)$. Frantzikinakis--Kuca conjecture. If $a$ is good for seminorm control along $\ell$-term arithmetic progressions for the system $(X,\mathcal X,\mu,T)$, then it is good for degree-$(\ell+1)$ seminorm control along $\ell$-term arithmetic progressions for this system. The paper states that this claim is beyond its scope and leaves it as an open question.