The tightly 3-interlacing conjecture for rank 2 modules

Fix integers $(k,n)$ with $k\ge 3$. Let $I$ and $J$ be tightly $3$-interlacing $k$-subsets, meaning that they are $3$-interlacing and $|I\cap J|=k-3$. Let $L(I,J)$ denote the rank $2$ module constructed from $I$ and $J$. Tightly 3-interlacing conjecture. The module $L(I,J)$ is rigid and indecomposable. The preceding results establish indecomposability and show that its associated dimension vector is a real root; rigidity is proved in the tame cases and remains conjectural in general.