No, it will not. The thermotime switch only receives power from the same circuit that causes the starter to crank over. It has no power other than that. The only thing that can really cause the cold start injector to stay open is if it physically sticks. If you disconnect the connector to the thermotime or to the cold start injector, the cold start injector should cease to function. If you want to find out if the cold start injector is stuck open or causing your rich condition, you need to disconnect the fuel line to it and plug it. This would also be a good time to 'search' the archives for rich running conditions.