this is correct, the lpg solenoid should only open if the engine control manager computer sees that operation is safe, if it does not meet all the criteria for safe operation, then it should shut down, or not open the fuel control solenoid.
in your case, my experience would say to replace all the relays in the relay box
The solenoid should open while cranking, with the key switch crank signal powering the LPG relay through a couple of other relays.
when you crank it, does the "hitacchi handset" show engine rpm in the 100 to 250 rpm range? (crank sensor and cam sensor check?)