T-structure criterion for CN-realizable matrices

Let $M$ be a strictly upper triangular $(0,2)$-matrix, meaning that its entries are $0$ or $2$, and suppose that for each $k\geq 3$ there is at most one pair $(i,j)$ with $j-i=k$ and $M(i,j)=2$. A graph on the grid alignment $G(M)$ has a T-structure when it satisfies conditions (C1), (C2), and (C3) specified in the source: path lengths satisfy (C1), and the edge configurations satisfy (C2) and (C3). The matrix $M$ is CN-realizable when it is the CN matrix of a pure braid projection. T-structure conjecture. If $M$ has a T-structure, then $M$ is CN-realizable. The source notes that some CN-realizable matrices have no T-structure, so the proposed condition is sufficient rather than necessary; its general status is not resolved.