Attainable profiles for conservation laws with flux function spatially discontinuous at a single point

Consider a scalar conservation law with discontinuous flux (1): \begin{equation*} \quad u_{t}+f(x,u)_{x}=0, \qquad f(x,u)= \begin{cases} f_l(u)\ &\text{if}\ x<0,\\ f_r(u)\ & \text{if} \ x>0, \end{cases} \quad \quad \quad(1) \end{equation*