The trace-positivity conjecture for words in positive definite matrices

Let $W$ be a word in two letters $A$ and $B$, and let $A$ and $B$ be real positive definite matrices. Trace-positivity conjecture. The word $W$ has positive trace for every pair $A,B$ if and only if $W$ is symmetric or a product of two symmetric words. The conjecture concerns characterizing precisely which words have universally positive trace; the paper later gives counterexamples to the broader positivity phenomenon for products of symmetric words in related constructions.