A Conquest does not require an SIC, it is a single pilot airplane. However, are you sitting right seat just to be sitting, or are you doing it because they need another pilot, say for insurance reasons???
Is the guy in the left seat going to let you fly? I know he may not right off the bat, but after a couple of flights once you have a feel for the plane, maybe you could sweet talk him into letting you fly (even if from the right seat) and then you can log PIC.
Another thing to think about, could this possible land you some flying time if the other pilot is not available, or on vacation or something? Even if you determine you can't log it all, it may be an advantage later on if you have the time to ride.