The integer-multiple criterion for early state exclusion in 7-qubit spin chains

Let $J$ be a $7 \times 7$ Jacobi matrix with symmetric spectrum that realizes perfect state transfer (PST). Its early state exclusion (ESE) property means that no state is excluded at an early time, as defined in the paper. Let the positive eigenvalues of $J$ be ordered, and let the smallest positive eigenvalue be denoted by $\lambda_{\min}$. Integer-multiple conjecture. $J$ does not have ESE if and only if every positive eigenvalue of $J$ is an integer multiple of $\lambda_{\min}$. This criterion is based on numerical experiments concerning ratios among the positive eigenvalues. Its status is not established by the supplied material, so it remains open pending a proof or counterexample.