Non-entropy shock comparison conjecture

Let $\mu\in\mathscr{N}_+$ and write $v:=\partial_x g:=\partial_x\mathsf{H}[\mu]$, where $v$ is a tractable solution. Non-entropy shock comparison conjecture. The measure $\mu$ restricted to the support of $\mathsf{M}_{\mathsf{non}}[g]$ equals $\mathsf{M}_{\mathsf{non}}[g]$:

$$\mu\lfloor_{\operatorname{supp}(\mathsf{M}_{\mathsf{non}}[g])}=\mathsf{M}_{\mathsf{non}}[g].$$

Equivalently, $\mu$ and $\mathsf{M}_{\mathsf{non}}[g]$ coincide on the non-entropy shocks of $v$. In particular, this would imply

$$I(\mathsf{e}_{\mu})\geq I(\mathsf{M}_{\mathsf{non}}[g])=\mathsf{Ent}_+(v).$$

The conjecture addresses the discrepancy between the rate function of the original measure and the entropy-production expressions associated with the non-entropy solution; examples in the paper show that the corresponding rate-function equalities do not hold in general.