The partition-model conjecture for optimally approximating exponential families

Let $N$ be the size of the finite sample space and let $k$ be the dimension of an exponential family. For a class $\mathcal{H}$ of exponential families, define

$$D_{N,k}(\mathcal{H})=\min\left\{\max D_{\mathcal{E}}:\mathcal{E}\in\mathcal{H}\text{ is an exponential family of dimension }k\text{ on }[N]\right\}.$$

In particular, let $D_{N,k}$ denote the corresponding quantity for the class of exponential families containing the uniform distribution. A partition-model conjecture.

$$D_{N,k}=\log\left\lceil\frac{N}{k+1}\right\rceil,$$

and the dimension-$D_{N,k}$-optimal exponential families containing the uniform distribution are partition models.

Partition models consist of probability measures whose restriction to each block of a fixed partition of the sample space is uniform. The conjecture identifies the optimal approximation error and asserts that all optimal families in the uniform-reference class have this structure.