The central-product Dehn function conjecture

Let $\mathfrak{k}$ and $\mathfrak{l}$ be nilpotent Lie algebras of step $k$ and $\ell$, respectively, with $k\geqslant \ell\geqslant 2$, and with one-dimensional centers $\mathfrak{z}$ and $\mathfrak{z}'$. Let $\theta: \mathfrak{z}\to \mathfrak{z}'$ be an isomorphism, and let $\mathfrak{k}\times_{\theta} \mathfrak{l}$ be their central product. Write $K$, $L$, and $G:=K\times_{\theta}L$ for the associated simply connected Lie groups. Central-product Dehn function conjecture. The Dehn function of $G$ satisfies

$$n^{k}\preccurlyeq \delta_G(n)\prec n^{k+1}.$$

This is proposed as a general picture for Dehn functions of central products. The supplied text does not state whether the claim has been proved or remains open.