The piecewise quasi-polynomial conjecture for lattice tetrahedra by multi-width

Let $\mathcal{T}_{w_1,w_2,w_3}$ denote the set of lattice tetrahedra of multi-width $(w_1,w_2,w_3)$, with

$$w_3\geq w_2\geq w_1>0.$$

The multi-width tetrahedron counting conjecture. There is a piecewise quasi-polynomial with four components counting lattice tetrahedra of multi-width $(w_1,w_2,w_3)$. There is one component for each combination of equalities among $w_3\geq w_2\geq w_1>0$. The leading coefficient when $w_3>w_2>w_1>0$ is twice the leading coefficient when $w_3=w_2>w_1>0$. For fixed $w_1$ and $w_2$, the value of $|\mathcal{T}_{w_1,w_2,w_3}|$ can take at most three values according as $w_3$ is odd, even, or equal to $w_2$. This is suggested by computed tables and by the preceding classifications, but remains unproved.