Discrete surface-area conjecture for simplices

Let $S$ be a $d$-simplex with the origin in its interior and with rational vertex directions. Let $\operatorname{Vol}_{\mathbb{Z}^d}(S)$ be normalized volume and let

$$\operatorname{Surf}_{\mathbb{Z}^d}(S)=\sum_{i=0}^d\operatorname{Vol}_{\pi_i(\mathbb{Z}^d)}(\pi_i(S)),$$

where $\pi_i$ is the linear projection vanishing at the $i$th vertex. The discrete surface-area conjecture.

$$\mu(S)\leq\frac12\frac{\operatorname{Surf}_{\mathbb{Z}^d}(S)}{\operatorname{Vol}_{\mathbb{Z}^d}(S)}.$$

This is a discrete analogue of Hadwiger's covering inequality and is unimodularly invariant. It holds in dimension two and up to a factor of two in arbitrary dimension, but remains open in full generality.