Gath's conjecture for Cygan--Korányi ball discrepancy

Let $q\ge3$, let $\mathbb H^q$ be the Heisenberg group, and let $\mathcal B_t=\delta_t\mathcal B_1$ be the Cygan--Korányi ball. Write the lattice-point error term as

$$\mathcal E_q(t):=\#\left(\mathbb Z^{2q+1}\cap\delta_t\mathcal B_1\right)-\operatorname{vol}(\mathcal B_1)t^{2q+2}.$$

Gath's conjecture. The optimal order of the error term is $2q-1$; that is, the infimum of the exponents $\kappa$ for which $|\mathcal E_q(t)|\lesssim t^\kappa$ holds equals $2q-1$. For $q=1$ the corresponding order is known to be optimal, while for $q\ge3$ the source gives an upper bound of order $t^{2q-\frac23}$, a lower bound of order $t^{2q-1}(\log t)^{\frac14}(\log\log t)^{\frac18}$, and a sharp second-moment estimate of magnitude $t^{2q-1}$. The conjecture therefore remains open in the stated range.