Nonconstant potential-field conjecture for strongly extendable perforated domains

Let $\boldsymbol{\mathrm{P}}\subset\Omega$ be a corresponding prototype with the strong extension property, and let $\mathcal{V}_{\mathrm{pot}}^{p}(\boldsymbol{\mathrm{P}})$ denote the $L^{p}$-closure of gradients of functions in $W^{1,p}(\boldsymbol{\mathrm{P}})$. Strong extension conjecture.

$$\mathbb{R}^{d}\cap\mathcal{V}_{\mathrm{pot}}^{p}(\boldsymbol{\mathrm{P}})=\emptyset.$$

This property is known when $\boldsymbol{\mathrm{P}}=\Omega$, but the authors cannot establish it for general $\boldsymbol{\mathrm{P}}$ and avoid using it in the remainder of the work. It is stated as non-essential except for uniqueness properties of the homogenized problem.