Nilmanifold characterization of good sequences for multiple convergence

Let $(s_n)$ be a sequence of integers. An $\ell$-step nilsystem is a measure-preserving system arising from a translation on a compact homogeneous space $G/\Gamma$, where $G$ is an $\ell$-step nilpotent Lie group and $\Gamma$ is a discrete cocompact subgroup. For $a\in G$, write $T_a(g\Gamma)=(ag)\Gamma$. Nilmanifold characterization conjecture. The following three statements are equivalent: (i) $(s_n)$ is good for $\ell$-convergence; (ii) $(s_n)$ is good for $\ell$-convergence for every $\ell$-step nilsystem; and (iii) for every $\ell$-step nilmanifold $X=G/\Gamma$, every $a\in G$, and every $f\in C(X)$, the sequence

$$\left(\frac{1}{N}\sum_{n=1}^N f(a^{s_n}\Gamma)\right)$$

converges as $N\to+\infty$. This would provide a convenient characterization of $\ell$-convergence, analogous to the spectral-theoretic characterization available for single mean convergence. The source gives no resolution for $\ell\geq 2$.