Conway–Sloane's non-isometry conjecture for tetralattice pairs

Let $L_{1;a,b,c,d}$ and $L_{2;a,b,c,d}$ be the two lattices in Conway and Sloane's real four-parameter family, with parameters $(a,b,c,d)\in\mathbb R^4$. Conway–Sloane's conjecture. For all real numbers $(a,b,c,d)\in\mathbb R^4$ satisfying

$$0<a<b<c<d,$$

the lattices $L_{1;a,b,c,d}$ and $L_{2;a,b,c,d}$ are isospectral but not isomorphic. The family gives pairs of isospectral lattices in Euclidean four-space; Conway and Sloane formulated the conjecture from computational evidence, while the paper studies its non-isometry assertion.