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Once you have the door open, you may need to spend some time at (1) in your picture until any roaches hanging out clear out a bit. This could take some effort. It may actually be important to get to this spot quickly (like before too many spawns take place), but I think you should be able to make forward progress. The cluster of queens at (2) will produce 8 new roaches every spawning cycle, which will take you 16 moves to clear while standing at (1). So you should be able to clear an additional 7 each spawning cycle, which is forward progress, so that's good.

That lone queen that starts in the middle of the room may cause some headaches, but I think she ends up in the eastern exit and so is only going to be producing three roaches per spawning cycle. Still manageable.

Also, note that the cluster of queens at (3) will eventually fill up the area, but none of them will approach you until you move out of the doorway past (4).

Eventually, you will get to a point where you don't have two roaches right next to the doorway at (4) and you can move into the doorway. This is where you want to be...then you should be in a position to kill roaches twice as fast, and you can start luring roaches out of the area filled up by the queens at (3).

At this point it's just a matter of getting rid of the excess. It may help to force some of the queens into different locations -- you can either do this by heading towards the exit on the east wall, which will force the queens at (2) to join the queens at (3), or you can move south and force the queens at (2) into the exit. This will severely impact their productivity, which is good for you.

This room is definitely a slog, and one that would benefit from a checkpoint or two. Persist!