Please don't get me wrong. I didn't feel any uncomfortable when answering your questions.
If Site A contains site B information and is Site A using https, the page will always show security warning.
For this situation, you cannot install the livesupport in Site B and add the cross domain script to site A.
You have two options:
1. Install the live support in each site. Don't use the cross domain solution.
2. If Site A is using https and Site B is not using https, install the live support in Site A instead of Site B.
Hope it helps.
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