Finiteness of even-dimensional thin simplices of lattice width at least two

A thin simplex is a lattice simplex of lattice width at most one. Finiteness conjecture. In each even dimension there are only finitely many thin simplices of lattice width $\geq 2$.

The claim concerns the apparent scarcity of thin simplices with lattice width at least two in even dimensions; the source provides computational motivation but no resolution.