How to properly learn when there is random classification noise?

The following problem is motivated by the one here from more than half a decade ago:

Let $$C$$ be a concept class that is efficiently proper PAC-learnable, i.e. there exists a learning algorithm that outputs $$h \in C$$, such that $$\mathrm{err}(h) \leq \epsilon$$, in addition to the usual PAC-guarantees. Suppose that this same class $$C$$ is PAC-learnable in the presence of random classification noise, but that the learning algorithm for $$C$$ in this setting. Show that $$C$$ is also proper PAC-learnable in the presence of random classification noise.

In the standard setting without noise, the existence of an efficient proper PAC-learning algorithm for the class $$C$$ implies that we can find a hypothesis $$h \in C$$ with an error rate of at most $$\epsilon$$. However, when introducing random classification noise, the challenge is that the labels of the instances may be flipped with some probability, say $$\eta$$, which complicates the learning process.

My initial approach was to adapt the standard learning algorithm for $$C$$ to account for the noise. I considered techniques such as noise-tolerant variants of the algorithm or methods to estimate and correct the flipped labels. However, I am uncertain about how to guarantee that the modified algorithm still outputs a hypothesis from the class $$C$$ (thus ensuring proper learning) while maintaining the error rate within the desired bounds. I'd appreciate any help.

PAC learnability is characterized (modulo very mild measure-theoretic assumptions, which we'll ignore here) by a finite VC-dimension. If you have a shattered set of size $$d$$ and a uniform distribution on those $$d$$ points, whenever your sample size $$n, you have no information about the labels of the remaining $$d-n$$ unseen points. Hence finite VC-dim is necessary even for realizable (noiseless) PAC.