Unique solvability of symmetric word equations

Let $A$ and $B$ be real positive definite matrices, and let a symmetric word equation be an equation obtained by evaluating a symmetric word in $A$ and $B$ at a prescribed matrix. The associated map is the map $h:A\mapsto S(A,B)$ from positive definite matrices to themselves, where $S(A,B)$ is the matrix represented by the symmetric word. Unique solvability conjecture. Every symmetric word equation is uniquely solvable. The preceding result establishes solvability for every symmetric word equation, while injectivity—and hence uniqueness—is stated here as believed but proved only in special cases.