If we started talking about what and how to write in tags, then by this point both Push and Hints have already worked. Fetch priority in html tags — too late feature for our discussion!
Waterfalls from my screenshots is formed from real browsers, this is not a simulation! So you can trust the data you see.
If what you see does not match your ideas, then it's time to correct them, and start believing what you saw on waterfalls.
Your site uses Push, as I see, but does not use all the advantages of HTTP2 (and 3) - try to decompose CSS and JS bundles, minify manually written JS code (click and scroll handlers, what's in the middle of the bundle) and also use variable font instead of four OpenSans files and you will get more more productivity.
And that's not all. To get more power from prefetch (instant), you must enable the browser cache for at least 1-3 minutes by parsing the expires headers from the server, and then the pages will open really instantly.
I hope you will use the tips and add speed to your site, and also make sure that I myself understand a little what I'm talking about ;-)
But I hope even more that LiteSpeed will give us Early Hints support!
Thank you for the discussion, we will be waiting for good news from LSWS and OLS!