These 005K's come from the same H-Factory that builds the 177, which has the same movement & same issues.
The regulating screw on the swanneck is indeed the culprit.
The screw has threads but the swanneck is not threaded on the inside....allowing the screw to fall out.
I had this happen to mine this week.
If you remove the caseback, you will most likely find the screw bound up in the gears of the movement below the ciruclar spring of the swanneck.
I used a small jeweler's screwdriver to move the gears slightly to unbind the screw and remove it.
I left the screw out, replaced the caseback and the watch runs perfectly.