The skin is unstable on first run after upgrade from 5.4.5 to a 6.2(.x)

Note that I don't remember exactly if we experienced this on 6.2 or 6.2.1.

To reproduce (I don't know if it will, because it seemed quite unstable):

• download and install 5.4.5 jetty-hsql version
  • no modification was made on the skin of this version
• upgrade the war to 6.2(.1) -> replace the webapps xwiki folder with the 6.2(.1) one
• start the wiki to have Distribution Wizard.
  • Problem 1: Distribution Wizard is displayed with Serif fonts, and a few layout issues. The layout is mostly ok (the display in a rectangular region, centered, etc) but the fonts are bad and inside the rectangle some layout is broken. No print screen of that, sorry.
• run through distribution wizard, finish the upgrade
• at the end of the distribution wizard, the wiki homepage is shown no skin at all is applied on the wiki (white page with text and links). No print screen of that, sorry.
• Managed to go to the preferences of the wiki, changed skin to colibri (or maybe to the skin page to change the base skin to colibri, I don't remember)
• Everything works fine on the main wiki
• Go to a subwiki, flamingo skin is applied on this wiki, but with issues in the top menu (sorry, no print screen here either)
  • we clicked around a few times (changed the page, refresh) and it started working out of the blue.
• went back on the main wiki (maybe navigated a bit more, I don't remember) and got the issue in the attached printscreen (top menu was all messy).
• I checked in the logs of the server (since I assumed that maybe less had issues being executed since part of the skin was fine and other part not), I couldn't see anything
• I didn't know what to do more, I restarted the wiki and everything went fine after.