Some shutters have a "pressed in " PC socket. The socket is a pressure fit into the case but with tight fitting cable and twisting motion they can come loose. The internal sync may be a simple contact that connects the outer sleeve to the center contact by mechanical action or contacts from the socket that make and break by mechanical action.

Your best bet is to take the shutter to a repair shop and let them replace the socket. The shutter should be modifiable to use the type shown on the B&H site if the original style is no longer available.