Thinning-invariant partition structures from stationary random measures

Let $\rho\trimmed$ be a stationary random measure on $\mathbb{R}$ with intensity-one asymptotic density, let $\mathscr{K}$ be the disjoint union of $\mathcal{M}_{+,1}(\mathbb{R})\times(0,1)$ and $1$, and let $\mathcal{M}_{\mathrm{st}}(\mathscr{K})$ be the measures invariant under shifts of the $\rho$ coordinate. For $Q\trimmed\this$ in $\mathcal{M}_{\mathrm{st}}(\mathscr{K})$, define $\mathcal{N}_Q$ by mixing the Poisson--Kingman structures $\nu_{(\Lambda_{\rho,m},0)}$ and the structure $\nu_{(0,1)}$. The thinning-invariant partition-structure conjecture. For any $Q,Q'\in\mathcal{M}_{\mathrm{st}}(\mathscr{K})$, if $\mathcal{N}_Q=\mathcal{N}_{Q'}$ then $Q=Q'$, and the set of thinning-invariant $\mu\in\mathcal{M}(\Delta_\infty)$ is precisely $\{\mathcal{N}_Q:Q\in\mathcal{M}_{\mathrm{st}}(\mathscr{K})\}$. This claims both uniqueness of the mixing measure and a complete classification of thinning-invariant partition structures; the supplied text gives no resolution status.