Localized test-function conjecture for adaptive transport discretizations

Let $\breve{\boldsymbol{U}}_{\tilde{\mathcal{T}}}^{\delta}$ be the discrete trial space on the refined partition $\tilde{\mathcal{T}}=\tilde{\mathcal{T}}(\mathcal{T},\overline{\mathcal{M}},r)$, let $\vartheta$ be the constant occurring in the estimator reduction argument, and let $c\in L_\infty(\Omega)$. For a given $F\in L_2(\Omega)$, consider a function $\tilde w$ whose support is restricted to the cells of $\mathcal{T}$ refined in $\tilde{\mathcal{T}}$. Localized test-function conjecture. There exist constants

$$\xi<\Big({\textstyle \frac{\sqrt{2}}{\vartheta}} \sqrt{(2\|c\|_{L_\infty(\Omega)}+1)^2+1}\Big)^{-1}$$

and $r\in\mathbb{N}$, such that there exists $(0,\tilde w)\in\breve{\mathbb{U}}_{\tilde{\mathcal{T}}}^{\delta}$ satisfying

$$\|\partial_{\mathbf b}\tilde w+c\tilde w-F\|_{L_2(\Omega)}\leq\xi\|F\|_{L_2(\Omega)},\qquad \|\tilde w\|_{L_2(\Omega)}\lesssim\|F\|_{L_2(\Omega)},$$

where $\tilde w$ vanishes outside the union of the cells of $\mathcal{T}$ that were refined in $\tilde{\mathcal{T}}$. Such a localized approximation property is the ingredient needed to complete the proof of the adaptive estimator-reduction theorem; the supplied text gives no resolution of the conjecture.