After a lot of trial and error, I've finally come up with an improved 5 shot cop strategy for FAITH: Chapter III speedruns. This strategy beats the old one in pretty much every way. The setup and execution is faster, the setup is more consistent and has less parts that can cause variance with both Jeffrey and cops behavior, and it also guarantees that Jeffrey gets caught on the table with no chance of slipping out. I start each demonstration by leaving the room to reset cops position to where he will be during runs and I also begin away from the wall to show that weird subpixel positioning will not ruin the strat.

Key things to note:

1. While approaching the room where you fight Jeffrey, make sure you hug the wall on the left. Upon transitioning, you want to make sure you are holding up left so that you immediately begin moving towards the left wall in the next room.

2. After you bump the left wall in the entryway of the clinic, begin moving up until Jeffrey spawns in and then stop moving. A good visual cue here is to look at John's feet and the top of the table. Jeffrey should spawn JUST before the bottom of his feet cross the top of the table. It's important that you don't stop early before he spawns or you may desync cop from where he needs to be to move consistently and execute the strat correctly.

3. When the front of Jeffrey's head crosses the pink line in the middle of the doorway up top, move down and face up. I like to get to where the top of John's head is parallel with the bottom edge of the tabletop, but I don't think it matters too much where you are as long as you aren't so high that Jeffrey can kill you as he rounds the corner.

EDIT: There is some room to modify and tweak this strategy at this point to your liking. Since the inception of this strategy, I have found even better consistency by moving down once Jeffrey moves around the top-right corner of the table, moving John down past the bottom edge of the tabletop and then turning around and waiting for Jeffrey to turn the top-left corner. This gives you some time and space to better prepare your crux timing.

4. Begin using crux right before Jeffrey begins to round the corner of the table.