The equality-case conjecture for maximal-volume clean lattice tetrahedra

Let $\mathscr{S}_k^3$ denote the family of clean lattice tetrahedra in $\mathbb{R}^3$ with $k$ interior lattice points, let $\operatorname{Vol}(T)$ denote the normalized volume, and let $T_{2k+1,4k+3,12k+8}$ be the tetrahedron specified by the source's notation. Equality-case conjecture. If $T\in \mathscr{S}_k^3$ and

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

then $T\simeq T_{2k+1,4k+3,12k+8}$. If $k>1$, then

$$\operatorname{int}(T)\cap\mathbb{Z}^3$$

is a set of collinear points whose line does not pass through any vertex of $T$. This conjecture predicts both the isomorphism type of every equality case and the configuration of its interior lattice points. The source presents it as a computationally motivated conjecture and gives no resolution.