CN-matrix conjecture for pure braid projections

Let $n$ be a positive integer. A braid projection is a braid diagram without over/under-crossing information. Its CN matrix $N(B)$ is the zero-diagonal matrix whose $(i,j)$ entry counts crossings between strands $s_i$ and $s_j$; it is therefore symmetric. A matrix is even when all its entries are even, and a zero-diagonal matrix is T0 if, whenever $1\leq i<j<k\leq n$, $M(i,j)=M(j,k)=0$ implies $M(i,k)=0$. CN-matrix conjecture. An $n\times n$ non-negative integer matrix $M$ is the CN matrix of some pure $n$-braid projection if and only if $M$ is an even T0 symmetric matrix. This conjecture is proposed alongside the related crossing-matrix and OU-matrix conjectures; no general resolution is given.