A Structure result for bricks in Heisenberg groups

We show that for a sufficiently big $\mathit{\text{brick}}$ $B$ of the $(2n+1)$-dimensional Heisenberg group $H_n$ over the finite field ${\mathbb{F}}_p$, the product set $B\cdot B$ contains at least $\mathrm{\mid }B\mathrm{\mid }\mathrm{/}p$ many cosets of some non trivial subgroup of $H_n$.