Egor, thanks for the reply, I was able to recreate the problem as you stated and correct it with the workaround you stated. Thanks for that, As I have had it happen several more times in the past two days and would just undo until it went away. I agree that it should show both width adjustments in the dialog, and I hope this is something that gets fixed.

Eric, thanks for sharing the file.
I made some tests and can confirm several things:
— This is not caused by Repeat, the behavior is the same without it.
— This happens for closed paths and not open ones.
— These are in fact two width markers.
— It’s easy enough to replicate from scratch with these steps:

1. Create an circle with a stroke

2. Grab Width Tool and modify the width of the ellipse in any place, effectively creating a width marker. Notice how the second marker automatically appears at the opposite side of the circle from this one, and one more marker at the rightmost point — this is where the path starts. This in fact is TWO markers, having the same width.

3. Grab the marker you created manually and drag it along the path until it hits the stop — this would be the first point of the closed path.

4. Grab the marker that used to be the opposite to the one you created and drag it into the same point.

Once you do that, these get 'glued' together. Now you have only two markers (one at the start and one at the end of the path, meeting in the single point), they have different widths, and any attempt to reposition one will create another marker, and double-clicking it gives an access to only one of two.

This is obviously stupid. The dialog at least should provide a way to access both — the same way it does when we move markers together within any other place on a path... See the GIF attached.
Notice how normally Illustrator avoids having only two markers for the stroke and keeps adding a third one once I try to delete the extras!

This should be fixed and I upvote the report.

But thankfully there is a simple enough workaround. See the second GIF.
1. Double-click the marker at the start/end of the path.
2. Copy the value.
3. Create a new marker with a double click from the side that is wrong.
4. Paste the copied width.
5. Merge this marker with the one you grabbed the width from.
6. Double-click again and enable the only option.
Now these two will change their width together if you try.

You don’t have to release the Repeat object even to do that.
Please comment back if it helps.

Egor, please find the shape in questions.

Eric, cant you share this piece, please? Just this one, in a separate file.