The nearly symmetric word conjecture for positive definite letters

Let $W$ be a word in two letters $A$ and $B$, evaluated at positive definite matrices. Call $W$ nearly symmetric when it has the structural symmetry defined in the paper.

Nearly symmetric word conjecture. A word has positive trace for every pair of positive definite letters if and only if the word is nearly symmetric.

The paper identifies a large class of words that guarantee positive eigenvalues and presents this conjecture as being supported by considerable evidence. The supplied text does not state that the conjecture has been resolved.