Advice for replacing wiring that runs through PVC piping...

So here's the setup, this is a small office (<50 network drops in the whole building) that has a small computer lab with 12 connections. Way before my time, these connections were run through PVC pipe in the floor that continues up a wall into the drop ceiling. Originally, these then connected to a pair of daisy-chained 10BaseT hubs with a single line running back to the actual network stack (after I discovered this gem, I replaced the hubs with a single switch that was lying around). My goal is to get all of these connections directly wired back to the network and remove the middleman if possible.

I'm currently running new cable and am wondering if anyone has tips for replacing that cable that's running through the PVC? I've tried just pulling it through from the end in the ceiling, but I think there are too many bends in the path so it won't budge. Opening up walls/floors is not an option, unfortunately.