The first-two-layers conjecture for spherical designs of Euclidean lattices

Let $\Lambda$ be a lattice, and let $M_1(\Lambda)$ and $M_2(\Lambda)$ denote its first and second layers. A finite subset of a sphere is a spherical $2$-design when it satisfies the defining moment conditions for a spherical $2$-design. First-two-layers conjecture. If the first two layers $M_1(\Lambda)$ and $M_2(\Lambda)$ of $\Lambda$ are spherical $2$-designs, then all the other layers are $2$-designs; in particular, $\Lambda$ is fully critical for the height $h$. The conjecture would extend the observed phenomenon that, for the strongly eutactic lattices examined by the authors up to dimension $8$, failure of full criticality is already detected in the second layer. It would also provide a sufficient criterion for full criticality beyond the condition established by the paper's theorem; the general assertion remains open in the supplied source.