On universal deformation rings and stable homogeneous tubes

Let k be a field of any characteristic and let $Λ$ be a finite dimensional k-algebra. We prove that if V is a finite dimensional right $Λ$-module that lies in the mouth of a stable homogeneous tube T of the Auslander-Reiten quiver $Λ$ with End_Λ(V) a division ring, then V has a versal deformation ring $R(Λ,V)$ isomorphic to k[\![t]\!]. As consequence we obtain that if k is algebraically closed, $Λ$ is a symmetric special biserial k-algebra and V is a band $Λ$-module with End_Λ(V) k that lies in the mouth of its homogeneous tube, then $R(Λ,V)$ is universal and isomorphic to k[\![t]\!].