3 problems

The $S_{\{1,j\}_n^2}$ conjecture. For sufficiently large $n$, the sets $S_{\{1,j\}_n^2}$ are not Laplacian realizable.

The $m=1$ non-realizability conjecture. For $n\geqslant3$, the set $S_{\{i,n\}_{n}^{1}}$ is not Laplacian realizable.

The authors' $S_{\{i,n\}_{n}^{m}}$-conjecture. For sufficiently large $n$ (in the stated formulation, $n\geqslant9$), no graph realizes $S_{\{i,n\}_{n}^{m}}$.