On universal deformation rings and stable homogeneous tubes

Let $\mathbf{k}$ be a field of any characteristic and let $\mathrm{\Lambda }$ be a finite dimensional $\mathbf{k}$-algebra. We prove that if $V$ is a finite dimensional right $\mathrm{\Lambda }$-module that lies in the mouth of a stable homogeneous tube $\mathfrak{T}$ of the Auslander-Reiten quiver $\mathrm{\Lambda }$ with ${\underset{‾}{\mathrm{E}\mathrm{n}\mathrm{d}}}_{\mathrm{\Lambda }}(V)$ a division ring, then $V$ has a versal deformation ring $R(\mathrm{\Lambda },V)$ isomorphic to $\mathbf{k}[\text{\hspace{0.17em}⁣}[t]\text{\hspace{0.17em}⁣}]$. As consequence we obtain that if $\mathbf{k}$ is algebraically closed, $\mathrm{\Lambda }$ is a symmetric special biserial $\mathbf{k}$-algebra and $V$ is a band $\mathrm{\Lambda }$-module with ${\underset{‾}{\mathrm{E}\mathrm{n}\mathrm{d}}}_{\mathrm{\Lambda }}(V)\cong \mathbf{k}$ that lies in the mouth of its homogeneous tube, then $R(\mathrm{\Lambda },V)$ is universal and isomorphic to $\mathbf{k}[\text{\hspace{0.17em}⁣}[t]\text{\hspace{0.17em}⁣}]$.