How to reproduce it:
1. Open the user's attached project "TimelineBug.zip"
2. Load the "SampleScene" Scene
3. Open the Timeline window
4. In the Hierarchy view, select a "Master" GameObject
5. Notice that in the Console window "StartPreview" message appears and clear the Console
6. Double-click on the "Nested" clip in the Timeline window > Clips view
7. Observe the Console window

Expected result: "StopPreview" and "StartPreview" messages appear in the Console window
Actual result: Only the "StopPreview" message appears in the Console window. IAnimationWindowPreview.StartPreview callback is not called

Reproducible with: 1.2.13 (2020.2.5f1), 1.6.0-pre.1 (2019.4.23f1, 2020.3.2f1, 2021.1.0f1, 2021.2.0a7)
Not reproducible with: 1.2.12 (2020.2.5f1)
Could not test with: 2018.4.33f1 - Timeline package is not introduced yet, and in-built Timeline doesn't have IAnimationWindowPreview

Notes:
- Preview button in the Timeline's toolbar has to be enabled