If $\tau_{1}\prec\tau_{2}$ are simplices of $T$, then $\mathfrak{d}^{\delta}$ is invariant with respect to $\widehat{\tau}_{1}-\widehat{\tau}_{2}$ on the region $\bigcup_{\theta\in[0,1-\delta]}\theta\widehat{\tau_{2}}+(1-\theta)\tau_{1}$. In particular, $\mathfrak{d}^{\delta}$ is constant equal to $w$ on the whole star of $w\in\operatorname{vert}(T)$ in $\operatorname{bsd}(T,T)$.