The flow $X:[0,T]\times \mathbb{R}\to\mathbb{R}$ defined by $X(t,x)= x+t\chi_{\mathbb{R}_+}(x)$, for all $(t,x)\in [0,T]\times \mathbb{R}$, is a regular Lagrangian flow solution to $(\star)$ in the sense of *Definition (4)* of the linked paper (for a.e. initial datum $x\in \mathbb{R}$, in fact for all, one has $(x+t\chi_{\mathbb{R}_+}(x))'=\chi_{\mathbb{R}_+}(x)=\chi_{\mathbb{R}_+}(x+t\chi_{\mathbb{R}_+}(x))$; moreover $X(t,\cdot)_\#\mathcal{L}^1\le\mathcal{L}^1$ for all $t\ge0$).

Note that according to the above definition, one can always modify a regular Lagrangian flow for a vector field $b$ at least on a countable number of flow lines (even to non-solution curves), always yielding to a regular Lagrangian solution; indeed the uniqueness has to be intended for the flow as an element of $L^\infty([0,T]\times \mathbb{R}^d,\mathbb{R}^d)$. The vector field $b$ itself is given as an element of $L^\infty([0,T]\times \mathbb{R}^d,\mathbb{R}^d)$.