Steps to reproduce:
1. Open the attached user's project "AtlasRepro.zip"
2. Ensure Sprite Packer is set to "Sprite Atlas V2" in Project Settings
3. Drag "Tex1.png" (set as Sprite) into "Assets/Pack"
4. Atlas updates correctly
5. Drag 'Tex2.png" (set as Default) into "Assets/Pack"
6. Atlas does not update which is expected
7. Set "Tex2.png" to Sprite
8. Atlas does not update which is not expected
9. Remove "Tex1" from "Assets/Pack"
10. Atlas updates to show "Tex2"
11. Add "Tex1" back to "Assets/Pack"
12. Atlas updates to remove "Tex2", and displays only "Tex1" which is not the expected result

Expected results: Sprite Packer re-packs folder when texture type is changed to sprite when inside a target folder
Actual Results: Sprite Packer does not re-pack folder when texture type is changed to sprite when inside a target folder

Reproducible with: 2021.3.24f1, 2022.2.17f1, 2023.1.0b13, 2023.2.0a13
Could not test on 2020.3.47f1 due to Sprite Atlas preview not being shown

Environment: Windows 10 22H2

Notes:
-Issue persists until a manual re-import of the Atlas asset. Restarting the Editor does not solve the issue.
-Video of the issue is attached (AtlasRepro.mp4)
-When targeting a folder with the sprite packer, the sprites inside are not packed when their texture type is changed from within the folder. When adding a texture that is not set to sprite, the texture seems to be cached and ignored.