If the APU is switched off but the APU bleed is on, will the engine bleed valve close automatically?
The answer is in the FCOM:

So, the ENG bleed valve closes if:

Reverse flow detected
ENG 1 starter valve not closed
APU bleed valve open
Bad things (Overpressure / Overheat / Leak)
ENG BLEED switch is off or FIRE pb pushed

So, if APU is not ON, the APU BLEED valve is closed, and then the condition 3 is false. If all the others conditions are also false, then the ENG valve would open.