The 9-pin socket should take a #4 screw with a small head. That's been true since the 7-pin came out in the 1940s.
This works out to 2.85mm. The #4 fit was always loose, I would think a 3mm screw would go in the hole.
> standard 9 pin tube sockets, like this one
Unfortunately the guys who make sockets don't use them. Often the threads fit but the head butts the side of the socket. When electronic parts were sold everywhere, it was easy to find small-head screws.
And looking at your picture, that socket does not have the dent next for the screw-head. There should be a good semi-circle, about a mm clearance all around the hole.
Model-shop for small screws.
Get the screw in crooked then tighten carefully to indent the shell for the screw head.