The conditional W-way MPST representation conjecture

Let $\widetilde D$ be a real-valued functional on bounded conditional $W$-tuples $(\boldsymbol{\pi}_{|G},p_G)$ over a fixed Polish alphabet pair $(\mathcal{X},\mathcal{G})$. Assume that $\widetilde D$ satisfies joint data-processing inequalities under main-system stochastic operators $T_{Y|XG}$ and conditioning-system stochastic operators $T_{H|G}$, is additive under tensor products of independent extensions, and vanishes on coincident tuples:

$$\widetilde D((\pi,\dots,\pi)_{|G}\,\|\,p_G)=0$$

for every $\pi$ and $p_G$. Let $\widehat{\mathcal{A}}$ be the unconditional four-stratum index space and set

$$\widehat{\mathcal{A}}_{\mathrm{cond}}=\widehat{\mathcal{A}}\times[1,\infty].$$

Conditional $W$-way MPST conjecture. There is a unique inner- and outer-regular finite Borel measure $\widetilde\mu$ on $\widehat{\mathcal{A}}_{\mathrm{cond}}$ such that

$$\widetilde D(\boldsymbol{\pi}_{|G}\,\|\,p_G)=\int_{\widehat{\mathcal{A}}_{\mathrm{cond}}}\mathsf{C}_{\alpha,\beta}(\boldsymbol{\pi}_{|G}\,\|\,p_G)\,d\widetilde\mu(\alpha,\beta),$$

and conversely every such measure defines a functional with the stated properties. The converse is established atom by atom, but the forward implication remains conjectural because the conditional spectral-exhaustion step is not proved; in particular, the conditional tropical and KL-edge factors remain part of the open step.