You would think but sadly, no.
They serve the actual tweet as "Cache-Control: no-cache, no-store, must-revalidate, pre-check=0, post-check=0"
https://pastee.org/bzeb5
EDIT: Just to check, I also looked at one of their css resources. https://si0.twimg.com/a/1330626450/t1/css/t1_core_logged_out...
Cache-Control: private, max-age=31449600
So I'm not confident Twitter actually understands how to set things.
PS: Hey Twitter, I can help =)