No, interlock switches only deal with ground side of starter-relay, which activates starter-solenoid. You need power going into junction-box in order for starter-relay to send power out to starter-solenoid.
Pull out voltmeter and trace starter power-circuit from start to end. Key ON, measure voltage at these locations:
1. battery's positive terminal, what is this voltage?
2. white wire coming out of starter-solenoid, what is this voltage?
3. white wire going into ignition switch, what is this voltage?
4. red/blue wire leaving ignition switch, what is this voltage?
5. red-blue wire entering junction-box, what is this voltage?
What voltage do you measure at all these locations?