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In 2010 Menon and Srinivasan published a conjecture for the statisticalstructure of solutions $\rho$ to scalar conservation laws with certain Markovinitial conditions, proposing a kinetic equation that should suffice todescribe $\rho(x,t)$ as a stochastic process in $x$ with $t$ fixed. In thisarticle we verify an analogue of the conjecture for initial conditions whichare bounded, monotone, and piecewise constant. Our argument uses a particlesystem representation of $\rho(x,t)$ over $0 \leq x \leq L$ for $L > 0$, with asuitable random boundary condition at $x = L$.

Scalar conservation laws with monotone pure-jump Markov initial conditions