The black hose is the drain for the cavity under the spoiler. And yes, I see the connector on the side of the motor, but I do not see the air line for the trunk lock in any pics -but this is certainly the area under the trim panel, underside of a Roadster trunk.

Problem is, if the top goes UP but not DOWN, the only sensors that are used ONLY for "down" are the top open and divider switches. He's already said he thinks they are bypassed, but unless you MEASURE That pin that I went to SO MUCH TROUBLE to describe, you do not know if the PTCM actuallys "sees" those switched closed. If it does not see it, then you get the symptom described.

You gotta MEASURE that pin with the engine running, if it is over about .6 volt, the top wont go down but it will go up.

I do not know what connector this is, in the photos, I don't have even a guess.