Seminorm-control conjecture for arithmetic progressions

Let $a\colon {\mathbb N}\to {\mathbb Z}$ be a strictly increasing sequence. For a system $(X,{\mathcal X},\mu,T)$, say that $a$ is good for seminorm control along $\ell$-term arithmetic progressions when the associated averages satisfy the relevant Gowers-Host-Kra seminorm estimate. Seminorm-control conjecture. If $a$ is good for seminorm control along $\ell$-term arithmetic progressions for $(X,{\mathcal X},\mu,T)$, then it is good for degree $\ell+1$ seminorm control along $\ell$-term arithmetic progressions for this system. The source explains that this is known for $\ell=1$ and that the general case would follow from a higher-order mixing statement for nilsystems; the full conjecture remains open.