Burnside conjecture for exceptional surface homeomorphism groups

Let $\Sigma$ be one of the four exceptional closed surfaces

$$\Sigma \in \{\mathbb{S}^2,T^2,\mathbb{R}P^2,K\}.$$

A subgroup is periodic if every one of its elements has finite order. Burnside conjecture for exceptional surfaces. For each $\Sigma \in \{\mathbb{S}^2,T^2,\mathbb{R}P^2,K\}$, every finitely generated periodic subgroup of $\operatorname{Homeo}(\Sigma)$ is finite. The sphere and torus cases are open without additional hypotheses, while the corresponding questions for the real projective plane and Klein bottle appear to be entirely open.