Globe artichoke seedlings question

My globe artichoke seedlings are doing very well but they have weak stems and lean this way and that. They are getting top heavy now.

Should I pot them on deeper to the level of the fist leaves, or will they man up on their own?

The seedlings don't look like the ones that readily make new roots along the stem such as toms and chillies, so I'm a bit worried that if I bury the stem in soil it might rot.

 

I only ever grew these once and if I remember correctly they are pretty unstable plants even when planted out and full grown.
Seem to remember them getting blown over very easily.
I'd be careful about burying the stems too deep, maybe some support would be a better idea.