Heisenberg area-distortion conjecture

Let $\mathcal{H}^3$ and $\mathcal{H}^5$ denote the three- and five-dimensional Heisenberg groups. The area distortion of $\mathcal{H}^3$ in $\mathcal{H}^5$ measures the filling area in $\mathcal{H}^3$ of loops whose filling area in $\mathcal{H}^5$ is bounded. Heisenberg area-distortion conjecture. The area distortion of $\mathcal{H}^3$ in $\mathcal{H}^5$ is $n^{3/2}$. The preceding discussion establishes lower and quadratic upper bounds, so the conjecture predicts the precise intermediate growth rate.