Are all VC-classes CPAC learnable?

Let $\mathcal{X}=\{0,1\}^*$ and let a (binary) classifier be a function $h:\mathcal{X}\to\{0,1\}$. Call $h$ computable if there exists a Turing machine that halts on every input $x\in\mathcal{X}$ and outputs $h(x)$ (i.e., $h$ is total computable). A concept class $\mathcal{C}$ is a set of computable classifiers; assume $\mathcal{C}$ is given effectively (e.g., by a computable enumeration of indices of total computable classifiers whose realized functions are exactly the elements of $\mathcal{C}$).

For a distribution $\mathcal{D}$ over $\mathcal{X}\times\{0,1\}$ and classifier $h$, define the risk $\mathrm{err}_{\mathcal{D}}(h)=\Pr_{(X,Y)\sim\mathcal{D}}[h(X)\neq Y]$. In the realizable setting, assume there exists a target concept $c\in\mathcal{C}$ such that $Y=c(X)$ almost surely under $\mathcal{D}$.

Define the VC dimension $\mathrm{VC}(\mathcal{C})$ in the standard way: it is the largest $d$ for which some $x_1,\dots,x_d\in\mathcal{X}$ are shattered by $\mathcal{C}$ (or $\infty$ if unbounded).

A CPAC (computable-PAC) learner for $\mathcal{C}$ is a single computable algorithm $A$ that, given rational $\epsilon,\delta\in(0,1)$ and an i.i.d. labeled sample $S=((x_1,y_1),\dots,(x_m,y_m))$ drawn from an unknown realizable distribution $\mathcal{D}$, outputs a finite description (e.g., an index) of a total computable classifier $h_S$ such that there exists a sample bound $m=m(\epsilon,\delta)$ for which

where the probability is over the draw of $S$. The learner is proper if it always outputs $h_S\in\mathcal{C}$, and improper otherwise (but still outputs a total computable classifier).