Ok, I'm telling you I've read the code.
Let's take these simple steps to get you where you wanna be:
1) On the stat screen that looks like this (that you say is changing), click the green watch to turn it off (in this case we'll act like Jennifer is wrong):
View attachment 1218463
Now the mini-stat screen will disappear.
2) Now, select the character card (pic) for this example in #1 below. Then (assuming we want Yumiko permanent) select her as in #2 below (don't worry about the green watch yet):
View attachment 1218467
Now the mini-stat screen disappears again.
3) Now the character card (pic) should look like this:
View attachment 1218470
4) Now, click on Yumiko's character card (pic) as in #1 below. Then finally click the green watch as in #2 below.
View attachment 1218471
5) Done. Now it doesn't matter what location or what characters are on main screen. Yumiko's mini-stat screen will stay until you mess w/ it.
I hope this clarifies it for you. This is exactly how the code is written to behave. If you don't wish it to behave this way that would be personal preference and not an error with the coding nor a bug. Regards.