Parametric mod-2 isoperimetric inequality for relative cycles

Let $F:X\rightarrow\mathcal{Z}_k([0,1]^n,\partial[0,1]^n;\mathbb{Z}_2)$ be a contractible family, and let $I_{k+1}([0,1]^n;\mathbb{Z}_2)$ denote the $(k+1)$-chains with mod-2 coefficients. Parametric mod-2 isoperimetric conjecture. There should exist a family $H:X\rightarrow I_{k+1}([0,1]^n;\mathbb{Z}_2)$ such that $\partial H(x)-F(x)$ is supported in $\partial[0,1]^n$ and

$$\operatorname{\textbf{M}}(H(x))\leq c(n)\max\left\{1,\operatorname{\textbf{M}}(F(x))^{\frac{n-k-1}{n-k}}\right\}.$$

The conjecture is motivated by the small-cube Federer--Fleming filling estimate and is posed specifically for mod-2 coefficients because analogous controlled parametric fillings can fail for integer coefficients.