Directed-hyperarea vector identity in $\mathbb{R}^d$

Suppose that a median-dual region is constructed around an interior point $\bm{p}_{j} \in \mathbb{R}^{d}$, and let the generic adjacent edge be $\bm{p}_{k} - \bm{p}_{j}$. For each $d$-simplex $T$ sharing this edge, let $\bm{n}_{j}^{T}$ be a normal vector associated with the $(d-1)$-facet of $T$ opposite $j$, with magnitude equal to the hypervolume of that facet:

$$\left\| \bm{n}_{j}^{T}\right\| = |\mathrm{opp}_{j}(T)|.$$

Directed-hyperarea vector identity. The directed-hyperarea vector satisfies

$$\bm{n}_{jk} = \frac{2}{d(d+1)} \sum_{T \in \left\{\mathbb{T}^{(d)}_{jk} \right\}} \bm{n}_{j}^{T}.$$

This identity is presented as a theoretical property governing median-dual regions in $\mathbb{R}^{d}$, relating the directed-hyperarea vector of an edge to the normal vectors of the opposite facets of all adjacent simplices. The supplied text does not indicate whether the identity has been proved or remains conjectural.