  We prove the existence and uniqueness of solutions to a class of stochastic
scalar conservation laws with joint space-time transport noise and
affine-linear noise driven by a geometric p-rough path. In particular,
stability of the solutions with respect to the driving rough path is obtained,
leading to a robust approach to stochastic scalar conservation laws. As
immediate corollaries we obtain support theorems, large deviation results and
the generation of a random dynamical system.
