Uniform integrability conjecture for graphs with smallest eigenvalue at least -3

Let $G$ be a connected graph, and let its smallest eigenvalue be denoted by $4\lambda_{\min}(G)4$. A graph is $\sigma$-integrable if its associated integral lattice admits an embedding, after scaling the bilinear form by $\sigma$, into a standard integer lattice. Uniform integrability conjecture. There exists a positive integer $\sigma$ such that any connected graph $G$ with smallest eigenvalue at least $-3$ is $\sigma$-integrable. This conjecture asks for a uniform integrability bound for all connected graphs whose smallest eigenvalue is at least $-3$; the preceding results establish integrability in important cases, but the existence of one universal positive integer remains open.