Asymptotic growth of averaged Dehn functions for nilpotent groups

It is proved that in any finite representation of any finitely generated nilpotent group of nilpotency class $l\geqslant1$, the averaged Dehn function $\sigma(n)$ is subasymptotic w.r.t. the function $n^{l+1}$. As a consequence it is stated that in every finite representation of a free nilpotent group of nilpotency class $l$ of finite rank $r\geqslant2$, the Dehn function $\sigma(n)$ is Gromov subasymptotic.