Llosa Isenrich–Pallier–Tessera conjecture on Dehn functions of central products

Let $K$ and $L$ be simply connected nilpotent Lie groups with one-dimensional centres, of nilpotency classes $k$ and $\ell$, respectively, and let their central product be

$$G=K\times_ZL.$$

Here $2\leqslant\ell\leqslant k$. Llosa Isenrich–Pallier–Tessera's conjecture. The Dehn function of $G$ satisfies

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

This proposes a sharp range for the Dehn function of central products of higher-class nilpotent groups, extending known quadratic upper bounds for central products of suitable 2-step nilpotent groups. The conjecture concerns the gap between the lower and upper polynomial exponents.