Boundary modification conjecture for localized sweepouts

Let $n>k\geq1$, let $F:X^p\rightarrow\mathcal{Z}_k([0,1]^n,\partial[0,1]^n;\mathbb{Z}_2)$ be a continuous family, and let $r\in(0,\tfrac12)$. There should exist a family $F':X\rightarrow I_k([r_0,1-r_0]^n;\mathbb{Z}_2)$ such that Boundary modification conjecture.

1. $F'(x)\llcorner(r,1-r)^n=F(x)\llcorner(r,1-r)^n$.
2. $\operatorname{\textbf{M}}(F'(x))\leq\operatorname{\textbf{M}}(F(x))+c(n)p^{\frac{n-k-1}{n-1}}$.
3. $\partial F'(x)$ is a continuous family of $(k-1)$-cycles in $\partial[r_0,1-r_0]^n$.
4. $\operatorname{\textbf{M}}(\partial F'(x))\leq c(n)\left(\frac{\operatorname{\textbf{M}}(F(x))}{r}+p^{\frac{n-k}{n-1}}\right)$.

The conjecture is motivated by the failure of a common coarea radius for arbitrary families and predicts that boundary modification costs only the scale of an optimal sweepout of the boundary.