[Blogging Prompts] Fix prompt card flickering on "My Site"

Expected behavior

There is no flickering on the prompt card while switching sites.

Actual behavior

There is some flickering on the prompt card while switching sites.

Steps to reproduce the behavior

1 - Install Jetpack and log in;
2 - Select a site that is a blog (which means the prompt card will be shown on "My Site" -> "HOME");
3 - Observe the dashboard: prompt card should be shown;
4 - Select a site that is not a blog (prompt card not shown);
5 - Select a site that is a blog and observe: there's some flickering with the prompt card (and also with the label next to the avatars after the card is loaded);
6 - Select a site that is not a blog again: there's some flickering that appears to happen when trying to hide the prompt card.

Tested on emulator, Android 12, WPAndroid 21.6

Start by reproducing the issue on the Android emulator described in the report, using WordPress Android 21.6, and switch between blog and non-blog sites from My Site. Trace the prompt card and adjacent avatar label during those transitions. Done means the prompt card and label switch without flickering when changing sites.