Lehmer-type shell design conjecture for two-dimensional Euclidean lattices

Let $L$ be a two-dimensional Euclidean lattice whose quadratic form is $ax^2+bxy+cy^2$. A nonempty shell of $L$ is the set of lattice vectors having a fixed nonzero norm, viewed on the corresponding sphere.

Shell design conjecture. The following assertions hold:

1. If $b^2-4ac=m^2(-3)$ for an integer $m$, then no nonempty shell of $L$ is a spherical $6$-design, and some nonempty shell is a spherical $4$-design. Moreover, if every nonempty shell of $L$ is a spherical $4$-design, then $b^2-4ac=-3$, so $L$ is an $A_2$-lattice.
2. If $b^2-4ac=m^2(-4)$ for an integer $m$, then no nonempty shell of $L$ is a spherical $4$-design, and some nonempty shell is a spherical $2$-design. Moreover, if every nonempty shell of $L$ is a spherical $2$-design, then $b^2-4ac=-4$, so $L$ is a $\mathbb{Z}^2$-lattice.
3. Otherwise, no nonempty shell of $L$ is a spherical $2$-design.

The conjecture seeks a complete classification of spherical designs arising from shells of two-dimensional Euclidean lattices. The paper presents it as a conjecture following computational observations; its resolution is not established in the supplied text.