02 Civic Manual 5th gear issue

Hello,
I have a honda civic 2002 lx that is a 5 spd manual. A few months ago, the gear shifter began to pop out when placed into fifth gear.
This took anywhere from 30 sec to a minute. This is the only gear this problem occurs in. I used to be able to hold it in and just keep my hand on it and I would feel it try to come in but now the shifter comes out of fifth gear the instant you put it in, along with some grinding noise.
I have heard that this is an issue with the synchros because they were made of brass and are flimsy. I have heard the solution is aftermarket carbon steel synchros that are really made for performance but outlast oem synchros. Is this true? I took it into a transmission shop and they want to rebuild the entire transmission, yet I have seen stories of people doing this even with the same exact car and problem that were almost new (20-30%2