Bridson's polynomial Dehn function conjecture for subgroups of products of free groups

Let $F_{m_1},\ldots,F_{m_n}$ be finitely generated free groups, and let $G$ be a finitely presented subgroup of their direct product. Its Dehn function measures the number of conjugates of defining relations needed to fill null-homotopic words of length at most $N$. Bridson's conjecture. The Dehn function of $G$ is polynomially bounded in $N$. This conjecture motivates the study of Dehn functions for subgroups of products of free groups; the paper proves polynomial upper bounds in important subclasses, but the general assertion remains open.