Characterization of crossing matrices of positive pure braids

Let $n\in\mathbb{N}$, and let $M$ be an $n\times n$ matrix. A positive pure braid is a pure braid admitting a diagram with only positive crossings, and its crossing matrix records the algebraic number of crossings between each pair of strands. A symmetric matrix is T0 when, for every $i<j$, $M(i,j)=0$ implies that there is some $k$ with $i<k<j$ such that $M(i,k)>0$ or $M(k,j)>0$.

Characterization conjecture. An $n\times n$ matrix $M$ is the crossing matrix of some positive pure braid if and only if $M$ is a non-negative integer T0 symmetric matrix.

The conjecture seeks a complete characterization of crossing matrices arising from positive pure braids, extending the known characterization for arbitrary pure braids by zero-diagonal integer symmetric matrices. Its resolution status is not specified in the supplied source context.