The surface-group sum-of-squares conjecture

Let $G$ be a surface group, and let $\mathbb{C}[G]^h$ denote the hermitian elements of its group algebra. Write $\Sigma^2\mathbb{C}[G]$ for the cone of finite sums of hermitian squares in $\mathbb{C}[G]$. Surface-group sum-of-squares conjecture. Every element $a\in\mathbb{C}[G]^h$ that is positive semidefinite in every finite-dimensional unitary representation of $G$ lies in

$$\Sigma^2\mathbb{C}[G].$$

The conjecture asks whether positivity on all finite-dimensional unitary representations is sufficient for a sum-of-squares certificate for surface groups. Surface groups are residually finite-dimensional, and the source notes that this may make the conjecture tractable; the general implication fails for groups containing $\mathbb{Z}^3$, while it is known for $\mathbb{Z}^2$.