The refinement-process obstruction to property (PM)

Let $\boldsymbol{a}$ and $\boldsymbol{b}$ be the vectors appearing in property $(PM)$, and let $\boldsymbol{ u}(t)$ denote a unit normal to a path in the refined mesh. Consider an open set $\Omega\subset\mathbb{R}^2$, a subdomain $\Omega'\subset\Omega$, and a refinement process with meshes $\mathcal{T}_{h_n}$. Suppose that for every segment $[b_0,b_1]\subset\Omega'$ with unit normal $\boldsymbol{n}$, every $V_0\in\mathcal{V}(b_0)$, every $V_1\in\mathcal{V}(b_1)$, every $V\in\mathcal{V}([b_0,b_1])$, and every $\varepsilon>0$, there is an $N\geq0$ such that for all $n\geq N$ there \exists a continuous union of edges $\gamma_n$ from $\mathcal{T}_{h_n}$ satisfying

\begin{array}{l} \boldsymbol{\gamma}_n(0)\in V_0,\\ \boldsymbol{\gamma}_n(1)\in V_1,\\ \boldsymbol{\gamma}_n(t)\in V\text{ for all }t, e\text{and } \int_0^1\bigl\lvert\boldsymbol{n}\cdot\boldsymbol{n}(t)\bigr\rvert\,dt\geq1-\varepsilon, \bigl. \text{Then there exists no }(\boldsymbol{a},\boldsymbol{b})\text{ such that property }(PM)\text{ is true for any }h_n.

The refinement-process obstruction to property (PM). Under the stated approximation condition, no fixed pair $(\boldsymbol{a},\boldsymbol{b})$ can make property $(PM)$ hold throughout the refinement sequence. The condition expresses that mesh-edge normals can approximate the normal of every segment sufficiently well along paths joining the prescribed neighborhoods. This is presented as a sufficient condition, while the source explicitly says that it is not known whether a refinement process satisfying the condition exists.