osr: chrome-runtime + alloy-style : focus is not fired in some scenarios

[Hethsron]

Describe the bug
Hello @magreenblatt

We recently encountered a problem with the focus in CEF. We had difficulty reproducing it because it rarely happened. But we're contacting you today because we've managed to find a simple scenario that you can easily reproduce.

To Reproduce
Steps to reproduce the behavior:

1. Run cefclient using the following command to to https://duckduckgo.com

cefclient.exe --use-alloy-style --enable-chrome-runtime --off-screen-rendering-enabled --enable-logging=stderr --v=1 --url=https://duckduckgo.com

2. Try to search something on the first page to be redirected on another.
3. Write something in the new input field and clear it.
4. Then click again in the input field, the carret, which originally appeared as a flashing vertical bar, is no longer displayed. But you can still type.

In our case, we observed this expected behaviour in a form created with VueJS. We had a drop-down list that wasn't displayed because the focus event hadn't been received, even though the activeElement property had changed.

We tested with Chrome and Edge and didn't have the problem. The problem only occurs when you activate the Chrome runtime with the Alloy style.

Expected behavior
We expect the behaviour to be in line with that of Chrome and Edge.

Video
https://github.com/user-attachments/assets/fbc9e37e-ce6e-4002-91ca-14be454da1fd

Impacted Version:

• OS: Windows 10, MacOS 13.2, Ubuntu 22.10
• CEF Version: 125, 126, 127, 128, 129, 130, 131, 132, 133 (using Chrome Runtime + Alloy Style)

Additional context
Yes, the problem is reproducible with cefclient as you can see.