The maximal-volume conjecture for clean lattice tetrahedra

Let $\mathscr{S}_k^3$ denote the family of clean lattice tetrahedra in $\mathbb{R}^3$ with $k$ interior lattice points, and let $\operatorname{Vol}(T)$ denote the normalized volume of $T$. Maximal-volume conjecture. If $T\in \mathscr{S}_k^3$, then

$$\operatorname{Vol}(T) \le \frac{1}{6!}(12k+8).$$

This conjecture proposes a linear upper bound for the volume of clean lattice tetrahedra in terms of their number of interior lattice points. The bound is motivated by computations; the source does not report a proof or a counterexample.