Boundary localization conjecture for sweepouts in domains with corners

Let $\Omega\subset\mathbb{R}^n$ be a connected domain with piecewise smooth boundary having $\theta$-corners, where $\theta\in(0,\pi)$. Fix $\eta>0$. For all $p\geq p_0(\Omega)$, $r\in(0,r_0(\Omega))$, and $\delta>0$, let $F:X^p\rightarrow\mathcal{Z}_k(\Omega,\partial\Omega;\mathbb{Z}_2)$ be a continuous map with no concentration of mass. Boundary localization conjecture. There should exist $F':X\rightarrow I_k(\operatorname{cl}(\Omega_r);\mathbb{Z}_2)$ such that

1. $\partial F'(x)$ is a continuous $\delta$-localized family in $\mathcal{Z}_{k-1}(\partial\Omega_r;\mathbb{Z}_2)$;
2. $\mathcal{F}(F(x)\llcorner\Omega_r,F'(x))<\eta$;
3. $\operatorname{\textbf{M}}(F'(x))\leq\operatorname{\textbf{M}}(F(x))+\frac{\operatorname{\textbf{M}}(F(x))}{r}p^{-\frac{1}{n-1}}+C(\Omega)\operatorname{\textbf{M}}(\partial\Omega)^{\frac{k}{n-1}}p^{\frac{n-k-1}{n-1}}$;
4. $\operatorname{\textbf{M}}(\partial F'(x))\leq c(\Omega)\left(\frac{\operatorname{\textbf{M}}(F(x))}{r}+p^{\frac{n-k}{n-1}}\right)$.

Moreover, if $F$ is a $p$-sweepout of $\Omega$, then $\partial F'$ is a $p$-sweepout of $\partial\Omega_r$ by $(k-1)$-cycles. This is the technical conjecture used in the Weyl-law proof; it permits a small interior perturbation and imposes no concentration of mass.