Small diameter multimedia cable?
It can be hard knowing what to search for in the first place if you don’t know the domain and the terminology.  This sounds like in-wall, so you’d want to search for “riser” or “horizontal” solid conductor cable.  There’s not much more to do than search for options and check specs to see what is thin enough.
I’ve run across bulk stranded cable as small as 3.75mm diameter, and since you have conduit you can technically run whatever you want in it, but just be aware that most RJ45 wall jacks are intended for use with solid conductors and punch down tools, so you’ll need to ensure you can properly terminate stranded wires.
Another thing that might help - a high quality cat5e or 6 will perform as well as a cat6a for short runs (~50-100ft maybe) since the spec is designed to account for longer runs. These tend to have smaller dividers or no divider in them at all, which would cut down the diameter. Belden 2413 is 5.3mm.
Hope this gives you some place to start your search!