Burillo's positive pure braid crossing-matrix conjecture

Let $n$ be a positive integer, and let $M$ be an $n\times n$ integer matrix. A zero-diagonal matrix is T0 if, whenever $1\leq i<j<k\leq n$, the conditions $M(i,j)=M(j,k)=0$ imply $M(i,k)=0$. A braid diagram is positive if all its crossings are positive, and $M$ is its crossing matrix when $M(i,j)$ is the number of positive crossings minus the number of negative crossings between strands $s_i$ and $s_j$ where $s_i$ is over $s_j$. Burillo's conjecture. $M$ is the crossing matrix of some positive pure $n$-braid diagram if and only if $M$ is a non-negative integer T0 symmetric matrix. The conjecture is known for $n\leq 5$, but remains open in general.