Steinke–Zakynthinou's linear CMI conjecture for realizable proper learning

Let $\mathcal H \subseteq \mathcal X\to\mathcal Y$ be a VC class of dimension $d$, and let $\mathcal A$ be a proper learning algorithm. For a distribution $\mathcal D$, write $\mathrm{CMI}_{\mathcal D}(\mathcal A_n)$ for the conditional mutual information of the algorithm on $n$ samples. Steinke–Zakynthinou's conjecture. There exists a real-valued function $f$ that is linear in $d$ and a constant $c\geq 0$ such that, for every nonnegative integer $d$ and every VC class $\mathcal H$ of dimension $d$, there is a proper learning algorithm satisfying, for every $n\geq d$, $\mathrm{CMI}_{\mathcal D}(\mathcal A_n)\leq f(d)$ for all distributions $\mathcal D$, and, for every realizable sample $s\in\mathcal Z^n$, $\mathbb E\hat R_s(\mathcal A_n(s))\leq c\,d/n$. This conjecture was refuted in the paper: the abstract states that the proper-learner CMI bound cannot be improved to $O(d)$, and the discussion explicitly says that the conjecture is refuted.