Conjecture on homological filling area and the Dehn function

Let $H$ be a finitely presented group. Denote by $\operatorname{FA}_H$ its homological filling-area function and by $\delta_H$ its Dehn function, and write $f\preceq g$ for the usual domination relation between filling functions. Homological filling-area conjecture. One has

$$\operatorname{FA}_H\preceq\delta_H.$$

The preceding result proves only the weaker bound using the superadditive closure of $\delta_H$. Thus this sharper comparison is presented as a suspicion in the source and remains open.