Contact your AHJ regarding use of neutrals at these switches.
You have wired this for an "old school" wiring plan. Changes were made in 2011 Code (which everyone has adopted) which require neutral in many, but not all switch locations, depending on where they are physically in the rooms. This may still be alright, but I would get the AHJ (your inspector) to sign-off ASAP. If the AHJ is displeased, you'll need to either wire certain segments with /4 cable or go to conduit; or change altogether to smart switches.
You are not allowed to use 2 cables to substitute for a /4 or /5 cable. You could if it was 5 volts DC, but it's mains AC power, and imbalanced currents in 2 cables causes big problems.
How to wire it old-school.
The problem with any 3-way/4-way complex is the dizzying re-use of the same colors for many tasks. Whereas it gets rather easy if you re-mark the wire colors by function. A 5-pack of colored electrical tape is $5. Always mark both ends of each wire the same.
Using this this color code: yellow are travelers, black is always-hot, red is switched-hot, and white is neutral. Here we go:
- on the leftmost cable, re-mark black and white wires yellow.
- on the center cable, re-mark black and red wires yellow.
- On the right cable, re-mark red and white wires yellow.
- On the lamp cable, re-mark the black wire red.
Remember when you re-mark a wire, it is now that color. I like to spiral the tape on so it marks the wire for much of its exposed length. The point of these colors is it makes the remaining wire the needed color.
Put your thinking cap on. Ready?
At 3-way switches, the 2 travelers go on the brass screws, and the remaining wire goes on black.
At 4-ways, the 2 travelers from one cable go on the brass screws, and the 2 travelers from the other cable go on the black screws.
At the wires which remain, join same colors with a wire nut.
Wow, that was not so bad. See what colors do for ya?