I think the message button shouldn’t be visible at all.
In desktop view you see this: open this in a new tab for desktop view
There is no message or chat button. It’s hidden by the component’s css:
But in mobile view, those are shown open this in a new tab for mobile view
The css is overwritten
So I think the problem is not that they do not work as expected, but that they aren’t hidden on mobile. You would not be able to click those buttons if they were hidden as expected.