Characterization of positivity for traces of 2-webs in the honeycomb graph

Let $H$ be the honeycomb graph, and let $A$ and $B$ be the matrices assigned to its edges, with $AB^{-1}$ defined. Two matrices are considered equivalent when they are related by gauge equivalence.

Positivity characterization conjecture. Traces of all $2$-webs in $H$ are positive if and only if, up to gauge equivalence, one of the following conditions holds:

\begin{enumerate} \item $A$ and $B$ are upper triangular with positive determinants; \item Among the three matrices $A$, $B$ and $AB^{-1}$, either two have positive eigenvalues and one has negative eigenvalues, or all three have negative eigenvalues. \end{enumerate}

This conjecture proposes that the two listed classes exhaust the matrix assignments producing positive traces for every $2$-web. The second class is established by the preceding theorem, while the upper-triangular case is another known source of positive traces; the claimed converse remains open.