Graphene do not support fairphone, but you can use calyxos, which supports fairphone (with verified boot?), and it is more up to date.

In general, I think calyx is probably more up-to-date and secure, with vanilla android experience. /e/ has its own unique athetics, and a SSO cloud service powered by nextcloud (last time I checked, nextcloud dont have good E2EE support, so I personally avoid putting my stuff on a nextcloud server hosted by others. But it is your choice)

Check DivestOS, it uses a lot of things from GrapheneOS + they have crrated their own security tools and applications.

I stopped using /e/OS last year and switched to CalyxOS.
The problem I had comes probably from the fact that /e/ tries to cover as many phones as possible (almost 230 as of now) without having a big enough testing pool, so stuff would break constantly even on the "officially supported" devices. And when you pointed out your problem on their forum, the answer you met was frequently something like "the problem is in your phone model, we can't do a lot about it".
Calyx instead supports only Pixel phones and a couple others, including the FP4, it's always tested for good (they have both a stable and a beta branch you can easily switch between) and their android and microG versions are usually more up to date.
Oh, and the Calyx community is one on the nicest and most helpful on the planet

I personally use calyx os and am pretty happy with it, they have a fairphone build

Love that there is a post about /e/OS!!!

After years of "degoogling" my phones, been rocking a Oneplus 6t (love this phone) with several OS's, but found /e/OS one night and haven't looked back.

I ran /e/os on my Fairphone 4 for a while. Wasn't a fan of the clearly iOS-inspired launcher or the outdated Android version, so I switched to Iodé, which delivers a more Android-y experience. The built-in, system-wide ad-blocker is great, but recently they added a "premium" model that allows for more rigorous blocking. I'm not sure how I feel about it.
You could also take a look at Ubuntu touch, which updated about a month ago, with improved support for the Fairphone 4.

In general I have the impression that GrapheneOS focuses more on security and a bit less on privacy.

Graphene only supports Pixel phones. You could try to port over the device tree, HALs, and build system changes from LineageOS, but you would have to maintain it all yourself and run your own builds. Im using a pixel and I really enjoy graphene, but I'm not sure if it's worth that much effort

I run /e/OS on my FP5. I like it a lot, I don't know much about the technology though so I can't tell it how it compares privacy wise. I am just happy that I get to run a phone without any Google/Apple involved. I use mostly f-droid apps and if I would really need something from the Play store I could access that through /e/'s App Lounge

Bit outside of the scope of your question, but CalyxOS is a graphene-like android distro but for more devices.

Isn’t graphene exclusive to pixel phones? (Phones by google)