Dehn functions of higher rank arithmetic groups of type $A_n$ in products of simple Lie groups

Suppose $\mathrm{\Gamma }$ is an arithmetic group defined over a global field $K$, that the $K$-type of $\mathrm{\Gamma }$ is $A_n$ with $n\ge 2$, and that the ambient semisimple group that contains $\mathrm{\Gamma }$ as a lattice has at least two noncocompact factors. We use results from Bestvina-Eskin-Wortman and Cornulier-Tessera to show that $\mathrm{\Gamma }$ has a polynomially bounded Dehn function.