Leibman's conjecture on polynomial subgroups of nilpotent Lie groups

Let $(X=G/\Gamma,\mu_X,T)$ be a connected nilsystem, with $T$ induced by left translation by $\tau\in G$. Write $G=G_0\Gamma$, where $G_0$ is the identity component, and write $\tau=\tau_0\gamma^{-1}$ with $\tau_0\in G_0$ and $\gamma\in\Gamma$. For $\alpha\in G_0$, define

$$g_\alpha(n)=(\alpha\gamma^{-1})^n\gamma^n.$$

For essentially distinct integral polynomials $p_0,\ldots,p_r$, let $\widehat{H}$ be the subgroup of $G^r$ defined in the source by the generated diagonal subgroup and the elements formed from the $g_\alpha(p_i(n))$. Leibman's conjecture. For any $d\in\mathbb{N}$,

$$\widehat{H}\cap (G_0)_d=\widehat{H}_d.$$

The conjecture is introduced as an ingredient that would imply the paper's other conjectures concerning correlation sequences and recurrence. It is attributed to Leibman, and the source does not state whether it has been resolved.