The simplex-only representation conjecture for symmetric W-way DPI-additive divergences

Let $W$ be a positive integer and let $D$ be a symmetric, $W$-way DPI-additive divergence on bounded $W$-tuples of probability distributions. Write

$$\Delta_W=\{\alpha\in[0,1]^W:\sum_{k=1}^W\alpha_k=1\}$$

and let $\mathsf{C}_{\alpha}$ denote the simplex-indexed divergence atom. A Borel measure is $\mathfrak{S}_W$-invariant when it is invariant under permutation of the $W$ coordinates. The simplex-only representation conjecture. Every such $D$ admits a representation

$$D(\boldsymbol{\pi})=\int_{\Delta_W}\mathsf{C}_{\alpha}(\boldsymbol{\pi})\,dm(\alpha),$$

for a finite, $\mathfrak{S}_W$-invariant Borel measure $m$ on $\Delta_W$. The claim is the simplex-restricted form of the proposed Choquet representation; the surrounding framework includes additional boundary strata, so whether all symmetric DPI-additive divergences are captured by simplex atoms alone is unresolved.