Certain graphs with exactly one irreducible $T$-module with endpoint $1$, which is thin: the pseudo-distance-regularized case

Let $\Gamma$ denote a finite, simple and connected graph. Fix a vertex $x$ of $\Gamma$ which is not a leaf and let $T=T(x)$ denote the Terwilliger algebra of $\Gamma$ with respect to $x$. Assume that the unique irreducible $T$-module with endpoint $0$ is thin, or equivalently that $\Gamma$ is pseudo-distance-regular around $x$. We consider the property that $\Gamma$ has, up to isomorphism, a unique irreducible $T$-module with endpoint $1$, and that this $T$-module is thin. The main result of the paper is a combinatorial characterization of this property.