When I wrote this, I hadn’t looked at the schematic and board layout documents. I now have.
On the schematic it shows that IREF pin 26 has been left unconnected. Try connecting a resistor of about 390k ohm to 680k ohm between IREF pin 26 and GND. I don’t know for sure, but because the current through this pin and resistor controls the brightness of the display, the OLEDs may not even light up without it.
Just to be safe, I’d connect a 10k ohm resistor between the ATmega328 RESET pin 29 (TP2) and VCC (TP7) to prevent any stray electrical noise from resetting the CPU.
Also, I’m pretty sure it doesn’t matter, but according to the documentation for the display, R/W pin 16, E/RD pin 17 and D3 to D7 pins 21 to 25 should all be connected to GND. D2 pin 20 should be left unconnected.