Boundary discrete surface-area conjecture for simplices

Let $S=\operatorname{conv}(\{v_0,\ldots,v_d\})$ be a $d$-simplex with $\mathbf{0}\in S\setminus\{v_0,\ldots,v_d\}$ and with rational vertex directions. For each $i$, let $\pi_i:\mathbb{R}^d\to\mathbb{R}^{d-1}$ be the linear projection vanishing at $v_i$, and let $I\subset\{0,\ldots,d\}$ be the set of labels of facets containing the origin. The boundary discrete surface-area conjecture.

$$\mu(S)\leq\frac12\frac{\sum_{i=0}^d\operatorname{Vol}_{\pi_i(\mathbb{Z}^d)}(\pi_i(S))+\sum_{i\in I}\operatorname{Vol}_{\pi_i(\mathbb{Z}^d)}(\pi_i(S))}{\operatorname{Vol}_{\mathbb{Z}^d}(S)}.$$

This extends the discrete surface-area conjecture from simplices with the origin in the interior to simplices with the origin on the boundary. The source proposes the extension but gives no general resolution.