How do I connect this PWM-controlled supply board properly?

[Ioannis]

... but there's some doubt if it will "remain ON".

I do not know what you mean by that. MosFets are not Thyristors or Triacs. A MosFet will be on if the drive is within specs and as long as there is drive. It will not remail ON if there is no drive.

Ioannis