OU-matrix conjecture for pure braids

Let $n$ be a positive integer. For an $n$-braid diagram, its OU matrix $U(b)$ is the non-negative integer zero-diagonal matrix whose $(i,j)$ entry counts crossings between strands $s_i$ and $s_j$ with $s_i$ over $s_j$. Let $M^T$ denote the transpose of $M$, and call a matrix even when all its entries are even. OU-matrix conjecture. An $n\times n$ non-negative integer matrix $M$ is the OU matrix of some pure $n$-braid diagram if and only if $M+M^T$ is an even T0 matrix. The OU matrix has been characterized for $n\leq 5$, while the conjecture remains open for general $n$.