The unique-solvability conjecture for symmetric word equations

Let $S(X,B_i)=P$ be a symmetric word equation, where $X$ is an $n\times n$ positive semidefinite matrix, the $B_i$ are fixed $n\times n$ positive definite matrices, and $P$ is an $n\times n$ positive semidefinite matrix. A solution is a positive semidefinite matrix $X$ satisfying the equation. Unique-solvability conjecture. Every symmetric word equation is uniquely solvable: for every choice of positive definite matrices $B_i$ and positive semidefinite matrix $P$, there is exactly one positive semidefinite solution $X$. Existence is known by the theorem of Hillar and Johnson; the conjecture concerns uniqueness.