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A circle domain $\Omega$ in the Riemann sphere is conformally rigid if everyconformal map of $\Omega$ onto another circle domain is the restriction of aM\"{o}bius transformation. We show that two rigidity conjectures of He andSchramm are in fact equivalent, at least for a large family of circle domains.The proof follows from a result on the removability of countable unions ofcertain conformally removable sets. We also introduce trans-quasiconformaldeformation of Schottky groups to prove that a circle domain is conformallyrigid if and only if it is quasiconformally rigid, thereby providing newevidence for the aforementioned conjectures.

Removability, rigidity of circle domains and Koebe's Conjecture