Parametric localized isoperimetric conjecture for domains with corners

Let $\Omega\subset\mathbb{R}^n$ be a connected domain whose piecewise smooth boundary has $\theta$-corners, with $\theta\in(0,\pi)$, and let $0\leq k<n$. Let $X^p$ be a $p$-dimensional parameter space. There should exist constants $c(\Omega)>0$ and $\delta(\Omega,L,p)>0$ such that, for every continuous $\delta(\Omega,L,p)$-localized contractible family $F:X^p\rightarrow\mathcal{Z}_k(\Omega,\partial\Omega;\mathbb{Z}_2)$ satisfying $\operatorname{\textbf{M}}(F(x))\leq L$, there is a map $H:X\rightarrow I_{k+1}(\Omega;\mathbb{Z}_2)$ with

$$\partial H(x)-F(x)\subset\partial\Omega$$

and

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

This is the technical form used in the paper's low-dimensional Weyl-law argument; it refines the preceding conjecture by incorporating localization, parameter dimension, and boundary regularity.