You’re going to have to take a closer look at the boxes with the 4-way switches.
In a more conventional wiring scheme, power is applied to one end and the load is either there as well or on the other end. In this configuration, it goes from a 3-way switch (three conductors plus a ground) to 4-way switch(es)(four conductors plus a ground to a 3-way switch. But in this configuration you are going to have more than one 3-wire Romex involved at the 3-way switches. By “involved” I mean more than one Romex at least partially connected to the 3-way switch.
Looking at your 3-way switches (and you should confirm) it’s pretty clear that all three conductors attached to each of the 3-way switches are all from one Romex. So what you don’t have if the more conventional wiring scheme I described above.
So that being the case, power is coming into one of your 4-way switch boxes. Going to copy/paste from another post to explain:
So if for both of your 3-ways you only have all three conductors from the SAME 3-wire Romex connected to them, then there are two possibilities.
1 - Power is being fed to the light first and then down to the 4-way switch. This will be a non-neutral at the switch box.
2 - Power is being fed to the 4-way switch box. This will be a neutral at the switch box.
The way you can differentiate these is by the number of 2-wire Romex “involved” at the 4-way switch box, related to this switch leg. By “involved” I mean either connected to the switch or routed to the other switches. Look at the 2 drawings below.
In the power to the switch box, there are two 2-wire Romex involved. There is one for the incoming power and one for the light. In the power to the light, there is only one 2-wire involved, the one to the light.
POWER TO SWITCH BOX
POWER TO THE LIGHT
You are going to have to figure out which of these you have. Once you know which of these you have will dictate how it’s wired.
This can be a bit tough to diagnose. If you can get to the light (hopefully only one) you can look to see if it’s just fed by a 2-wire Romex or if power is coming into the light first. If it’s multiple lights or they are high hats, then it’s a mess.
Hopefully this is enough to get you started.
We’ll give this more thought.