Realizable approximate ERM with constant conditional mutual information

Let $\mathcal{X}$ be a domain, let $\mathcal{W}$ be a class of functions $f:\mathcal{X}\to\{0,1\}$ with VC dimension $d\geq 1$, and let $n\geq 1$. For a dataset $(x,y)\in(\mathcal{X}\times\{0,1\})^n$, write $\ell(h,(x,y))$ for the empirical $0$-$1$ loss, and let $\mathsf{CMI}(A)$ denote the conditional mutual information of an algorithm's output. Realizable CMI conjecture. There exist absolute constants $c,c'$ such that, for every such $\mathcal{W}$ and $n$, there is a randomized or deterministic algorithm $A:(\mathcal{X}\times\{0,1\})^n\to\mathcal{W}$ satisfying

$$\mathsf{CMI}(A)\leq c d$$

and, for every $(x,y)\in(\mathcal{X}\times\{0,1\})^n$, if some $h\in\mathcal{W}$ has $\ell(h,(x,y))=0$, then

$$\mathbb{E}[\ell(A(x,y),(x,y))]\leq c'\frac{d}{n}.$$

This is the realizable setting, in which a consistent hypothesis exists. The conjecture seeks the sharper $d/n$ empirical-error rate while keeping conditional mutual information linear in VC dimension; the source presents it as a potentially easier alternative to the agnostic conjecture.