Universal deformation rings of string modules over a certain symmetric special biserial algebra

Let $\text{k}$ be an algebraically closed field, let $\text{A}$ be a finite dimensional $\text{k}$-algebra and let $V$ be a $\text{A}$-module with stable endomorphism ring isomorphic to $\text{k}$. If $\text{A}$ is self-injective then $V$ has a universal deformation ring $R(\text{A},V)$, which is a complete local commutative Noetherian $\text{k}$-algebra with residue field $\text{k}$. Moreover, if $\mathrm{\Lambda }$ is also a Frobenius $\text{k}$-algebra then $R(\text{A},V)$ is stable under syzygies. We use these facts to determine the universal deformation rings of string $\text{Ar}$-modules whose stable endomorphism ring isomorphic to $\text{k}$, where $\text{Ar}$ is a symmetric special biserial $\text{k}$-algebra that has quiver with relations depending on the four parameters $\bar{r}=(r_0,r_1,r_2,k)$ with $r_0,r_1,r_2\ge 2$ and $k\geq 1$.