The following VC dimension bound was established in [this MO post][1],
$$
\mathrm{VCdim}(H) \le 2\cdot \mathrm{VCdim}(\mathrm{SG}(F)),
$$
where $\mathrm{SG}(F) := \{\{x \in X \mid f(x) \le 0\}\}$ is the *subgraph* of $F$.

On the other, hand it's well-known that Rademacher complexity can be bounded via VC dimension like so.
$$
\mathbb E\, R_n(H) \lesssim\sqrt{\dfrac{\mathrm{VCdim}(H)}{n}}.
$$

We conclude that
$$
\mathbb E\,R_n(H) \lesssim \sqrt{\frac{\mathrm{VCdim}(\mathrm{SG}(F))}{n}}.
$$

In particular, because $\mathrm{VCdim}(\mathrm{SG}(F_{\mathrm{lin}})) = d$, we conclude that

$$
\mathbb E\,\widetilde R_n(F_{\mathrm{lin}}) \lesssim \sqrt{\frac{d}{n}}.
$$


  [1]: https://mathoverflow.net/a/420482/78539