Is there a simplicial set $X$ in which there is an edge $(f\colon C\to D)\in X_1$ (i.e., $d_0(f)=D$ and $d_1(f)=C$) and a 2-simplex $\tau\in X_2$ such that $$d_0(\tau)=s_0(D),\quad d_1(\tau)=f, \quad d_2(\tau)=f\qquad (\ast)$$ and $\tau$ is nondegenerate?
I observed that the degenerate simplex of the form $\tau = s_1(f)$ has the property $(\ast)$, and my question is kind of asking about the converse. (One approach of showing that there isn't such an example would be to let $\tau$ be a 2-simplex with $(\ast)$, and then to show that $\tau=s_1(f)$. But just using the simplicial identities this seems pretty hopeless.)
Note that nerves of categories can't provide such an example. But if there is a singular simplicial set for which this is true, I prefer to be given such an example.