Are you saying on top of this wiring scheme, I should also be connecting Pin 4 (GND) of J6 of each module together as well?
Yes, but I left an important bit out. If you just connect them together you will create a ground loop via the signal shields. This may make things worse. Maybe Hypex could (indeed should) provide the definitive answer. But at it core, whenever I see a single wire with no apparent reference/ground return it sets off alarm bells. Control of return current is critical in getting the best performance and is one of the things that makes or breaks the PC and internal layout of almost any design. The return current path for PS enable on the second and third amps is the common ground of all the amplifier modules. Which is currently the chassis via the signal shield. Since it is a balanced input this isn't the disaster it would be it if were an unbalanced input, but when we are eking out the last slivers of performance, it is always important to be clear what is happening. It depends on how the PCB is laid out.
There is a basic principle - every wire and every PCB trace is a resistor. Any current flowing in any wire or trace results in a voltage drop. Ground references can cease to be clean, as can any other part of the circuit. A ground loop is a particularly evil variant of this, as any loop is an inductor, and any nearby oscillating magnetic field will induce current in the loop. Current flowing in any loop that has been created results in a voltage drop around the loop, and the typical result is mains hum appearing. Even the ground traces in an amplifier module can (and usually do) have a voltage drop across them. It depends where on the PCB you are picking up the reference. My worry is that the signal shield is going to an assumed quiet ground in the PCB. If it is, and there it has become the return path for the PS enable signal, that relatively dirty standby power may add noise to the amp as it returns through the quiet part. This may not actually be an issue. It is quite possible that the signal shield is tied to the same bit of PCB as the standby power ground pin is. In which case it matters little. Hypex should provide guidance. Indeed they should provide clear guidance on how multiple modules can be set up and interconnected in a single chassis.
OK, after writing the above I did a bit of a search, and they do provide most of the guidance, and happily, they design their amplifiers the right way. So they don't have some notion of an internal quiet ground. Yay!
https://www.hypex.nl/img/upload/doc/an_wp/AN_Legacy_pin_1_problems.pdf
The take away? The current layout is probably OK.
There is no clear defined external ground for the amplifier modules - currently it is via the signal shield. Since this is the ground return for PS enable, this may not be optimal, but given Hypex's documentation is less likely to cause issues than it might with other designs. The optimal answer would be to use a multi-throw relay, and individually switch each amp using its own standby PS. But that may be unnecessary overkill. I would check with Hypex on what the recommended wiring is. They will probably say that what you have is the best answer. But it is an evil and vexed problem in general, and the right answer is not always the obvious one.