Search bar on qt.io broken when focused via keyboard

This is about the search bar on doc.qt.io and www.qt.io.

How to reproduce:

• focus the search bar using the keyboard (hit tab multiple times, maybe after clicking above the search bar to reduce the amount of tab hits)
• enter a search term
• hit enter

What happens:

• the page reloads with the search term in the URL but no search results are displayed

What should happen:

• search results should be displayed

The search bar only becomes functional after mouse-clicking in it.
I've attached a video that demonstrates the issue.

I can reproduce this on Linux and Windows with Firefox, Chrome and Edge.

Focusing the search bar with the keyboard seems like a overly tedious way to start a search, and I'd agree if I wasn't using the popular Firefox extension Vimium that allows me to hit gi to focus the first visible input field. It's working perfectly for literally every other website I'm using.