Bridson's polynomial Dehn function conjecture for finitely presented residually free groups

A group $G$ is residually free if, for every $g\in G\setminus\{1\}$, there is a homomorphism $\phi\colon G\to F_2$ to the rank-two free group such that $\phi(g)\neq 1$. For a finitely presented group, 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. Every finitely presented residually free group has polynomially bounded Dehn function. The conjecture is motivated by structural similarities between residually free groups and subgroups of direct products of free groups. Its general status remains open; the source notes in particular the longstanding open problem of determining the precise Dehn functions of the Stallings--Bieri groups.