A family of geometric quantum states for the Affleck-Kennedy-Lieb-Tasaki (AKLT) system

I am not a physicist. My motivation comes from generalizing a geometry problem due to the late Sir Michael Atiyah using graphs and tensor networks.

Given a finite graph (let's say simple, though this is not necessary), assume that at each vertex, you have a number of spin 1/2 quantum states equal to the valence of that vertex, 1 state for each incident edge. If for each edge, you first skew-symmetrize the two spin 1/2 quantum states at each end of the edge, and then you symmetrize the spin 1/2 states at each vertex, you obtain the so called VBS state (unless I am misunderstanding the definition, in which case please correct me). VBS stands for Valence Bond Solid. This occurs in the context of an Affleck-Kennedy-Lieb-Tasaki (AKLT) model in condensed matter physics.

In my construction, the quantum states I am interested in are constructed as follows:

step 1: the quantum state at vertex $i$ corresponding to an edge joining vertex $i$ to vertex $j$ is the +1 eigenvector of the operator $v_{ij}.\sigma$, where $v_{ij}$ is the direction from the point at vertex $i$ to the point at vertex $j$ and $\sigma$ is the $3$-vector of Pauli matrices, and

step 2: at a vertex $i$, one then symmetrizes all the spin 1/2 states corresponding to all the edges that are adjacent to $i$. One then obtains a state in the $d_i/2$ spin representation, where $d_i$ is the valence of the vertex $i$ with respect to the graph.

step 3: the "geometric" quantum state is then the tensor product over the set of the graph vertices of the symmetric tensor products constructed in step 2.

This defines a family of geometric quantum states parametrized by mapping the set of vertices of the graph to $\mathbb{R}^3$, with the condition that neighboring vertices are mapped to distinct points in $\mathbb{R}^3$. I will call this family the family of geometric quantum states of an AKLT system.

I have conjectured that any geometric quantum state in the previously defined family has a nonzero overlap with the VBS state. Note that the latter represents the quantum ground state of an AKLT system.

My motivation was mathematical. I have just figured out how to physically interpret my construction. My question is: from a physical point of view, is such a family of geometric quantum states of any interest?