FX Carbon sleeve liner loose fit- fix with heavy grease?

I think that in time that will be bad. That grease is going to ooze and make a mess. Just my opinion.

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Agreed. The CS is actually normally glued/epoxied to the liner, not the shroud. The CS stiffens the liner enough that the minor play should not be a hinderance. If you think that it's too much, contact UA and ask them what needs done. They may ask for a return to see what the issue is.
mike

Grease will move so bad idea, just epoxy it or at the very least locktite it.

I wrap the entire liner length with a thin teflon tape (I guess the blue cover) and that made a tighter fit. I was always against to glue=bond the liner to the CF tube.

If you want "the carbon fiber...cool", some grease will be just fine.
If you want the little extra barrel stiffness that the carbon fiber tube will help with, then YES...coat the entire barrel with a light coat of slow setting, quality two part epoxy and slide the two together. Obviously wiping away the squeeze-out as you go.
Use alcohol and q-tips to get in close near where the barrel will enter the receiver. Make it clean and free of epoxy while it's still wet.
Let it stand vertical, and unsupported (not leaning on the carbon fiber), overnight to fully set for the best results.

Any non-epoxy methods, will NOT provide the extra barrel stiffness that many desire.

Mike