Before I get into details, there is always the possibility of a poor connection, solder or tube socket or corroded tube pin.
That aside, PB is right. The reason I want more data is a suspicion that we have acquired some bad, possibly fake, 431s - there are a LOT of fake everythings coming out of [country names withheld], not just Rolexes. We seem to have had this problem only recently, or at least mostly.
A proper 431 should meet the specification. The most complete specification is that for the ON semiconductor product, which is what I used in the original design. It includes a very complete description of the noise character, which is better than most of these "adjustable Zener equivalent" chips - that's why we use that chip. You can swap the 431s to check - swap the connection from 12AT7 pins 3 and 8 to their respective kreg connections, at one end or the other. If the noise moves then it's the 431.
If you don't want to unsolder the 431 and replace it (I wouldn't want to, my hands are too shaky these days) you can replace the chip with a resistor as described in the referenced thread. I'm not checking that thread while posting this, so let us know if it's unclear. The C4S current is about 3.8mA and the chip drops 2.5 volts, so the perfect resistor is about 660 ohms; 680 ohms is close enough and widely available. This will be quieter than even the best 431, if it is metal film or wirewound or bulk metal foil. It raises the driver's effective plate impedance by about mu times the resistance; mu is nominally 60 so about 55K total. The 2A3 Miller capacitance is about 70pF giving a corner frequency of 41kHz. At 20kHz that 1dB reduction, so probably not audible unless you have bat ears.
I think that same thread mentions substituting an LED for the resistor. The problem is, we don't know much about LEDs other than the one we use which is noted for both low noise and a low dynamic impedance. That LED, the HLMP-6000, runs about 1.57 volts, significantly less than the target 2.5 volts. It should work, we've used it in previous incarnations of this circuit, but I moved to 2.5 volts for a reason, I think it's a more reliably good sounding operating point, given the variety of 12AT7s out there. (This has to do with differences in grid-cathode contact potential, if anyone cares.) My compromise would be to use an HLMP-6000 in series with a 240 ohm resistor, which moves the corner frequency to over 100kHz.