The two-by-two unique-solvability conjecture for symmetric word equations

Let $S(X,B)=P$ be a symmetric word equation in positive definite matrices, with $X$, $B$, and $P$ of size $2\times2$. Two-by-two unique-solvability conjecture. Symmetric word equations in $2\times2$ positive definite matrices have unique solutions. The paper proves uniqueness for one particular word equation before proposing this conjectural generalization; no resolution of the general two-by-two claim is given in the supplied text.