On skew Laurent polynomial rings over locally nilpotent rings

Let R be a locally nilpotent ring. Let σ be an endomorphism of R and let δ be a σ-derivation of R. Then the skew Laurent polynomial ring R[X,X−1;σ,δ] does not contain a non-zero idempotent. This answers a question posed by Greenfeld, Smoktunowicz and Ziembowski.