Minimality conjecture for initial threshold graphs generated by RFI

Let $N>0$ and $r\geq 2$ be fixed, and let $I=(0,N]$. Let $T_G(a_1,\ldots,a_r)$ be the initial threshold graph obtained from RFI$(N,r)$. Minimality conjecture. The graph $G$ is minimal: if $\widetilde{G}\preceq G$ is also $I$-eigenvalue free, then $\widetilde{G}=G$. This conjecture asserts that no smaller threshold graph is $(0,N]$-eigenvalue free. The authors report exhaustive computational searches for several positive values of $N$ that found no smaller example, but the general claim remains open.