On universal deformation rings and stable homogeneous tubes

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