Equidistribution of dilated polynomial curves in nilmanifolds

Let $X=N/\Gamma$ be a compact nilmanifold, where $N$ is a nilpotent Lie group admitting dilations and $\Gamma$ is a cocompact discrete subgroup. Let $p$ be a polynomial curve in $N$. The horizontal torus of $X$ is $N/[N,N]\Gamma$.

Equidistribution conjecture. The canonical projections onto $X$ of the dilations of $p$ equidistribute if and only if their projections onto the horizontal torus of $X$ equidistribute.

The theorem preceding this conjecture establishes the analogous equivalence for Heisenberg manifolds. The conjecture proposes that the same phenomenon holds for every compact nilmanifold whose covering nilpotent Lie group admits dilations.