Strict hierarchy conjecture for step-$I_0$ sets

A set $E\subset\mathbb{N}$ is called $k$-step-$I_0$ if it has the interpolation property for $k$-step nilsequences. Strict hierarchy conjecture. For every $k\in\mathbb{N}$, there exists a set that is $(k+1)$-step-$I_0$ but not $k$-step-$I_0$. The preceding proposition establishes the first nontrivial case: a $2$-step-$I_0$ set that is not $1$-step-$I_0$; the conjecture asks whether the hierarchy remains strict at every step.